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R0047-24136 · q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0011
Use prime=37, original a=11. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).
G001 — Use prime=37, original a=11. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [eulercrit-domain-qualification]
After
derived = [[prime, a, normalized_class, odd_prime, nonzero_class, certificate]]
Conditions: none
List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [37]
After
derived = [[[2, 1], [3, 1], [4, 1], [5, 2], [6, 1]], Yes]
Conditions: none
A composite has a factor at most its square root;complete trial division qualifies the modulus,not a guessed square/Euler equality.
Rule: Euler_modulus_primality_complete_trial · Uses: G001, S001
Before
Euler ORIGINAL signed quotient times modulus := mul(0, 37)
After
Euler ORIGINAL signed quotient times modulus = 0
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Euler ORIGINAL signed a minus modulus multiple := sub(11, 0)
After
Euler ORIGINAL signed a minus modulus multiple = 11
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [11, 37, 0]
After
derived = [11, Yes]
Conditions: none
Use a=qp+r with0<=r<p,including negative a;retain original a where the interface requests it.
Rule: Euler_Euclidean_class_before_hypothesis · Uses: G001, S004
Before
source roles = [37, 11, 11]
After
derived = [Yes, Yes]
Conditions: none
Qualify odd primality and the nonzero class first.Define Legendre by the square table before comparing a power.
Rule: Euler_odd_prime_nonzero_qualified · Uses: G001, S005
Before
Euler square root candidate1 := mul(1, 1)
After
Euler square root candidate1 = 1
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Euler square root candidate2 := mul(2, 2)
After
Euler square root candidate2 = 4
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Euler square root candidate3 := mul(3, 3)
After
Euler square root candidate3 = 9
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Euler square root candidate4 := mul(4, 4)
After
Euler square root candidate4 = 16
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Euler square root candidate5 := mul(5, 5)
After
Euler square root candidate5 = 25
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
Euler square root candidate6 := mul(6, 6)
After
Euler square root candidate6 = 36
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
Euler square root candidate7 := mul(7, 7)
After
Euler square root candidate7 = 49
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
Euler square root candidate8 := mul(8, 8)
After
Euler square root candidate8 = 64
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
Euler square root candidate9 := mul(9, 9)
After
Euler square root candidate9 = 81
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
Euler square root candidate10 := mul(10, 10)
After
Euler square root candidate10 = 100
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
Euler square root candidate11 := mul(11, 11)
After
Euler square root candidate11 = 121
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
Euler square root candidate12 := mul(12, 12)
After
Euler square root candidate12 = 144
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Euler square root candidate13 := mul(13, 13)
After
Euler square root candidate13 = 169
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
Euler square root candidate14 := mul(14, 14)
After
Euler square root candidate14 = 196
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
Euler square root candidate15 := mul(15, 15)
After
Euler square root candidate15 = 225
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
Euler square root candidate16 := mul(16, 16)
After
Euler square root candidate16 = 256
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
Euler square root candidate17 := mul(17, 17)
After
Euler square root candidate17 = 289
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
Euler square root candidate18 := mul(18, 18)
After
Euler square root candidate18 = 324
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
Euler square root candidate19 := mul(19, 19)
After
Euler square root candidate19 = 361
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
Euler square root candidate20 := mul(20, 20)
After
Euler square root candidate20 = 400
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
Euler square root candidate21 := mul(21, 21)
After
Euler square root candidate21 = 441
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
Euler square root candidate22 := mul(22, 22)
After
Euler square root candidate22 = 484
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
Euler square root candidate23 := mul(23, 23)
After
Euler square root candidate23 = 529
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
Euler square root candidate24 := mul(24, 24)
After
Euler square root candidate24 = 576
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
Euler square root candidate25 := mul(25, 25)
After
Euler square root candidate25 = 625
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
Euler square root candidate26 := mul(26, 26)
After
Euler square root candidate26 = 676
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
Euler square root candidate27 := mul(27, 27)
After
Euler square root candidate27 = 729
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
Euler square root candidate28 := mul(28, 28)
After
Euler square root candidate28 = 784
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
Euler square root candidate29 := mul(29, 29)
After
Euler square root candidate29 = 841
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
Euler square root candidate30 := mul(30, 30)
After
Euler square root candidate30 = 900
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
Euler square root candidate31 := mul(31, 31)
After
Euler square root candidate31 = 961
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
Euler square root candidate32 := mul(32, 32)
After
Euler square root candidate32 = 1024
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
Euler square root candidate33 := mul(33, 33)
After
Euler square root candidate33 = 1089
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
Euler square root candidate34 := mul(34, 34)
After
Euler square root candidate34 = 1156
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
Euler square root candidate35 := mul(35, 35)
After
Euler square root candidate35 = 1225
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
Euler square root candidate36 := mul(36, 36)
After
Euler square root candidate36 = 1296
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
source roles = [37, 11]
After
derived = [[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 12], [8, 27], [9, 7], [10, 26], [11, 10], [12, 33], [13, 21], [14, 11], [15, 3], [16, 34], [17, 30], [18, 28], [19, 28], [20, 30], [21, 34], [22, 3], [23, 11], [24, 21], [25, 33], [26, 10], [27, 26], [28, 7], [29, 27], [30, 12], [31, 36], [32, 25], [33, 16], [34, 9], [35, 4], [36, 1]], [14, 23], 1]
Conditions: none
Enumerate every nonzero square;roots give+1 and complete absence gives−1.This route does not use Euler criterion to compute Legendre.
Rule: Euler_ALL_nonzero_square_table_defines_Legendre · Uses: G001, S042
Before
Euler binary power square0 := mul(11, 11)
After
Euler binary power square0 = 121
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
source roles = [0, [1, 11, 18], 37, 11, 18]
After
derived = [null, 121, 1, 10, 9]
Conditions: none
Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation.
Rule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S044
Before
Euler binary power accumulate1 := mul(1, 10)
After
Euler binary power accumulate1 = 10
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
Euler binary power square1 := mul(10, 10)
After
Euler binary power square1 = 100
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
source roles = [1, [1, 10, 9], 37, 11, 18]
After
derived = [10, 100, 10, 26, 4]
Conditions: none
Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation.
Rule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S047
Before
Euler binary power square2 := mul(26, 26)
After
Euler binary power square2 = 676
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
source roles = [2, [10, 26, 4], 37, 11, 18]
After
derived = [null, 676, 10, 10, 2]
Conditions: none
Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation.
Rule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S049
Before
Euler binary power square3 := mul(10, 10)
After
Euler binary power square3 = 100
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
source roles = [3, [10, 10, 2], 37, 11, 18]
After
derived = [null, 100, 10, 26, 1]
Conditions: none
Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation.
Rule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S051
Before
Euler binary power accumulate4 := mul(10, 26)
After
Euler binary power accumulate4 = 260
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
Euler binary power square4 := mul(26, 26)
After
Euler binary power square4 = 676
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
source roles = [4, [10, 26, 1], 37, 11, 18]
After
derived = [260, 676, 1, 10, 0]
Conditions: none
Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation.
Rule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S054
Before
source roles = [37, [14, 23], 1, 1]
After
derived = [1, Yes, one finite instance is not a general theorem proof]
Conditions: none
p−1 represents−1 modulo p,not integer equality.Compare the two computations for this instance;one finite table is not a general theorem proof.
Rule: Euler_decode_p_minus1_and_instance_only_comparison · Uses: G001, S055
S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S054 · premise_and_result_audit — Recompute Euler binary power square4 with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute Euler binary power accumulate4 with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S051 · premise_and_result_audit — Recompute Euler binary power square3 with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S049 · premise_and_result_audit — Recompute Euler binary power square2 with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S047 · premise_and_result_audit — Recompute Euler binary power square1 with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute Euler binary power accumulate1 with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Recompute Euler binary power square0 with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S042 · premise_and_result_audit — Recompute Euler square root candidate36 with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute Euler square root candidate35 with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute Euler square root candidate34 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute Euler square root candidate33 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute Euler square root candidate32 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute Euler square root candidate31 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute Euler square root candidate30 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute Euler square root candidate29 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute Euler square root candidate28 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute Euler square root candidate27 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute Euler square root candidate26 with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute Euler square root candidate25 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute Euler square root candidate24 with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute Euler square root candidate23 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute Euler square root candidate22 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute Euler square root candidate21 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute Euler square root candidate20 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute Euler square root candidate19 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute Euler square root candidate18 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute Euler square root candidate17 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute Euler square root candidate16 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute Euler square root candidate15 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute Euler square root candidate14 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute Euler square root candidate13 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute Euler square root candidate12 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute Euler square root candidate11 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute Euler square root candidate10 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute Euler square root candidate9 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute Euler square root candidate8 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Euler square root candidate7 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute Euler square root candidate6 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute Euler square root candidate5 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Euler square root candidate4 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Euler square root candidate3 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Euler square root candidate2 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Euler square root candidate1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Recompute Euler ORIGINAL signed a minus modulus multiple with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed quotient times modulus with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(37, 11, 11, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 2206c3f8dddd4a787e7ec5c73cc694d26871cd68ec59c345b9cc17df4cdac54ccandidate record SHA-256: 3c60b9cae211dc971a710ddb4d7d6cdc5bdfe5dee88986d50cb8109ca0576189semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED