Prime-Field Characteristic, Multiplicative Powers, Affine Equations and Frobenius
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261031-r06-finite-field-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Characteristic Is the Additive Order of One
Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(2, [0, 1, 0], 2, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (2, [0, 1, 0], 2, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
2. Characteristic Is the Additive Order of One
Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(3, [0, 1, 2, 0], 3, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (3, [0, 1, 2, 0], 3, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
3. Characteristic Is the Additive Order of One
Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(5, [0, 1, 2, 3, 4, 0], 5, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 0], 5, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
4. Characteristic Is the Additive Order of One
Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(7, [0, 1, 2, 3, 4, 5, 6, 0], 7, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (7, [0, 1, 2, 3, 4, 5, 6, 0], 7, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
5. Characteristic Is the Additive Order of One
Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
6. Characteristic Is the Additive Order of One
Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
7. Characteristic Is the Additive Order of One
Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], 17, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], 17, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
8. Characteristic Is the Additive Order of One
Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], 19, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], 19, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
9. Characteristic Is the Additive Order of One
Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(23, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], 23, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (23, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], 23, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
10. Characteristic Is the Additive Order of One
Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(29, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 0], 29, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (29, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 0], 29, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
11. Characteristic Is the Additive Order of One
Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(31, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0], 31, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (31, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0], 31, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
12. Characteristic Is the Additive Order of One
Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(37, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0], 37, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (37, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0], 37, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
13. Characteristic Is the Additive Order of One
Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(41, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 0], 41, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (41, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 0], 41, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
14. Characteristic Is the Additive Order of One
Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(43, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0], 43, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (43, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0], 43, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
15. Characteristic Is the Additive Order of One
Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(47, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 0], 47, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (47, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 0], 47, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
16. Characteristic Is the Additive Order of One
Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(53, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 0], 53, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (53, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 0], 53, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
17. Characteristic Is the Additive Order of One
Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(59, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 0], 59, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (59, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 0], 59, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
18. Characteristic Is the Additive Order of One
Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(61, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 0], 61, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (61, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 0], 61, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
19. Characteristic Is the Additive Order of One
Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(67, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 0], 67, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (67, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 0], 67, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
20. Characteristic Is the Additive Order of One
Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0], 71, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0], 71, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
21. Characteristic Is the Additive Order of One
Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
22. Characteristic Is the Additive Order of One
Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(79, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 0], 79, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (79, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 0], 79, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
23. Characteristic Is the Additive Order of One
Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(83, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 0], 83, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (83, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 0], 83, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
24. Characteristic Is the Additive Order of One
Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(89, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 0], 89, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (89, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 0], 89, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
25. Characteristic Is the Additive Order of One
Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.
View answer and explanation
(97, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 0], 97, additive_order_of_one)
The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (97, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 0], 97, additive_order_of_one).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-characteristic · candidate
26. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=1. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(1, [1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (1, [1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
27. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=2. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(2, [2, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (2, [2, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
28. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=3. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(3, [3, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (3, [3, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
29. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=4. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(4, [4, 2, 1, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (4, [4, 2, 1, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
30. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=5. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(5, [5, 3, 4, 9, 1, 5, 3, 4, 9, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (5, [5, 3, 4, 9, 1, 5, 3, 4, 9, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
31. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=6. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(6, [6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (6, [6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
32. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=7. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(7, [7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (7, [7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
33. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=8. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
34. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=9. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(9, [9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1, 9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (9, [9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1, 9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
35. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=10. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(10, [10, 13, 14, 24, 8, 22, 17, 25, 18, 6, 2, 20, 26, 28, 19, 16, 15, 5, 21, 7, 12, 4, 11, 23, 27, 9, 3, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (10, [10, 13, 14, 24, 8, 22, 17, 25, 18, 6, 2, 20, 26, 28, 19, 16, 15, 5, 21, 7, 12, 4, 11, 23, 27, 9, 3, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
36. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=11. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(11, [11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (11, [11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
37. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=12. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(12, [12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (12, [12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
38. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=13. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(13, [13, 5, 24, 25, 38, 2, 26, 10, 7, 9, 35, 4, 11, 20, 14, 18, 29, 8, 22, 40, 28, 36, 17, 16, 3, 39, 15, 31, 34, 32, 6, 37, 30, 21, 27, 23, 12, 33, 19, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (13, [13, 5, 24, 25, 38, 2, 26, 10, 7, 9, 35, 4, 11, 20, 14, 18, 29, 8, 22, 40, 28, 36, 17, 16, 3, 39, 15, 31, 34, 32, 6, 37, 30, 21, 27, 23, 12, 33, 19, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
39. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=14. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(14, [14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1, 14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (14, [14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1, 14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
40. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=15. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(15, [15, 37, 38, 6, 43, 34, 40, 36, 23, 16, 5, 28, 44, 2, 30, 27, 29, 12, 39, 21, 33, 25, 46, 32, 10, 9, 41, 4, 13, 7, 11, 24, 31, 42, 19, 3, 45, 17, 20, 18, 35, 8, 26, 14, 22, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (15, [15, 37, 38, 6, 43, 34, 40, 36, 23, 16, 5, 28, 44, 2, 30, 27, 29, 12, 39, 21, 33, 25, 46, 32, 10, 9, 41, 4, 13, 7, 11, 24, 31, 42, 19, 3, 45, 17, 20, 18, 35, 8, 26, 14, 22, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
41. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=16. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(16, [16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (16, [16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
42. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=17. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(17, [17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1, 17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (17, [17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1, 17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
43. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=18. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(18, [18, 19, 37, 56, 32, 27, 59, 25, 23, 48, 10, 58, 7, 4, 11, 15, 26, 41, 6, 47, 53, 39, 31, 9, 40, 49, 28, 16, 44, 60, 43, 42, 24, 5, 29, 34, 2, 36, 38, 13, 51, 3, 54, 57, 50, 46, 35, 20, 55, 14, 8, 22, 30, 52, 21, 12, 33, 45, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (18, [18, 19, 37, 56, 32, 27, 59, 25, 23, 48, 10, 58, 7, 4, 11, 15, 26, 41, 6, 47, 53, 39, 31, 9, 40, 49, 28, 16, 44, 60, 43, 42, 24, 5, 29, 34, 2, 36, 38, 13, 51, 3, 54, 57, 50, 46, 35, 20, 55, 14, 8, 22, 30, 52, 21, 12, 33, 45, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
44. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=19. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(19, [19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1, 19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (19, [19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1, 19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
45. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=20. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
46. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=21. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
47. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=22. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(22, [22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (22, [22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
48. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=23. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(23, [23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1, 23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (23, [23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1, 23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
49. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=24. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
50. The Nonzero Multiplicative Group Gives a p−1 Power Identity
Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=25. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.
View answer and explanation
(25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1], 1, Yes, nonzero_multiplicative_group_cycle)
Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1], 1, Yes, nonzero_multiplicative_group_cycle).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-power-cycle · candidate
51. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=1, b=1, c=1; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(1, 1, 1, 1, 0, 1, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (1, 1, 1, 1, 0, 1, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
52. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=2, b=2, c=0; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(2, 2, 0, 2, 2, 0, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (2, 2, 0, 2, 2, 0, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
53. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=3, b=3, c=0; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(3, 3, 0, 2, 4, 0, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (3, 3, 0, 2, 4, 0, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
54. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=4, b=4, c=0; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(4, 4, 0, 2, 6, 0, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (4, 4, 0, 2, 6, 0, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
55. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=5, b=5, c=9; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(5, 5, 9, 9, 3, 9, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (5, 5, 9, 9, 3, 9, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
56. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=6, b=6, c=11; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(6, 6, 11, 11, 3, 11, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (6, 6, 11, 11, 3, 11, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
57. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=7, b=7, c=13; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(7, 7, 13, 5, 13, 13, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (7, 7, 13, 5, 13, 13, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
58. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=8, b=8, c=15; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(8, 8, 15, 12, 8, 15, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (8, 8, 15, 12, 8, 15, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
59. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=9, b=9, c=17; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(9, 9, 17, 18, 6, 17, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (9, 9, 17, 18, 6, 17, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
60. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=10, b=10, c=19; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(10, 10, 19, 3, 27, 19, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (10, 10, 19, 3, 27, 19, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
61. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=11, b=11, c=21; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(11, 11, 21, 17, 15, 21, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (11, 11, 21, 17, 15, 21, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
62. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=12, b=12, c=23; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(12, 12, 23, 34, 4, 23, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (12, 12, 23, 34, 4, 23, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
63. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=13, b=13, c=25; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(13, 13, 25, 19, 23, 25, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (13, 13, 25, 19, 23, 25, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
64. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=14, b=14, c=27; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(14, 14, 27, 40, 4, 27, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (14, 14, 27, 40, 4, 27, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
65. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=15, b=15, c=29; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(15, 15, 29, 22, 26, 29, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (15, 15, 29, 22, 26, 29, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
66. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=16, b=16, c=31; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(16, 16, 31, 10, 44, 31, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (16, 16, 31, 10, 44, 31, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
67. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=17, b=17, c=33; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(17, 17, 33, 7, 53, 33, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (17, 17, 33, 7, 53, 33, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
68. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=18, b=18, c=35; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(18, 18, 35, 17, 45, 35, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (18, 18, 35, 17, 45, 35, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
69. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=19, b=19, c=37; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(19, 19, 37, 60, 8, 37, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (19, 19, 37, 60, 8, 37, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
70. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=20, b=20, c=39; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(20, 20, 39, 32, 40, 39, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (20, 20, 39, 32, 40, 39, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
71. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=21, b=21, c=41; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(21, 21, 41, 7, 67, 41, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (21, 21, 41, 7, 67, 41, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
72. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=22, b=22, c=43; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(22, 22, 43, 18, 62, 43, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (22, 22, 43, 18, 62, 43, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
73. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=23, b=23, c=45; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(23, 23, 45, 65, 19, 45, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (23, 23, 45, 65, 19, 45, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
74. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=24, b=24, c=47; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(24, 24, 47, 26, 64, 47, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (24, 24, 47, 26, 64, 47, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
75. Solve a Nonzero-Coefficient Affine Equation in the Prime Field
Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=25, b=25, c=49; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
View answer and explanation
(25, 25, 49, 66, 32, 49, Yes)
Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (25, 25, 49, 66, 32, 49, Yes).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-affine-solve · candidate
76. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=0, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(0, 2, 0, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (0, 2, 0, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
77. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=1, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(1, 3, 1, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (1, 3, 1, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
78. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=2, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(2, 5, 2, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (2, 5, 2, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
79. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=3, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(3, 7, 3, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (3, 7, 3, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
80. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=4, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(4, 11, 4, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (4, 11, 4, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
81. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=5, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(5, 13, 5, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (5, 13, 5, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
82. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=6, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(6, 17, 6, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (6, 17, 6, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
83. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=7, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(7, 19, 7, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (7, 19, 7, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
84. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=8, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(8, 23, 8, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (8, 23, 8, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
85. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=9, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(9, 29, 9, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (9, 29, 9, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
86. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=10, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(10, 31, 10, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (10, 31, 10, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
87. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=11, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(11, 37, 11, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (11, 37, 11, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
88. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=12, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(12, 41, 12, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (12, 41, 12, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
89. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=13, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(13, 43, 13, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (13, 43, 13, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
90. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=14, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(14, 47, 14, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (14, 47, 14, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
91. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=15, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(15, 53, 15, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (15, 53, 15, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
92. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=16, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(16, 59, 16, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (16, 59, 16, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
93. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=17, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(17, 61, 17, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (17, 61, 17, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
94. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=18, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(18, 67, 18, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (18, 67, 18, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
95. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=19, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(19, 71, 19, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (19, 71, 19, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
96. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=20, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(20, 73, 20, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (20, 73, 20, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
97. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=21, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(21, 79, 21, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (21, 79, 21, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
98. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=22, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(22, 83, 22, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (22, 83, 22, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
99. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=23, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(23, 89, 23, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (23, 89, 23, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
100. Frobenius Fixes Every Element of This Prime Field
Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=24, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
View answer and explanation
(24, 97, 24, Yes, Frobenius_on_prime_field)
Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (24, 97, 24, Yes, Frobenius_on_prime_field).
Reviewed step-by-step derivation
Finite field · qf-mod31-fielddeep-frobenius · candidate
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