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R0047-24154 · q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0029
Use odd prime p=11, original a=4. Give (prime, normalized_class, nonzero_roots, Legendre_symbol, modular_power, signed_power, equal, certificate).
G001 — Use odd prime p=11, original a=4. Give (prime, normalized_class, nonzero_roots, Legendre_symbol, modular_power, signed_power, equal, 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-independent-comparison]
After
derived = [[prime, normalized_class, nonzero_roots, Legendre_symbol, modular_power, signed_power, equal, 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 = [11]
After
derived = [[[2, 1], [3, 2]], 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, 11)
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(4, 0)
After
Euler ORIGINAL signed a minus modulus multiple = 4
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 = [4, 11, 0]
After
derived = [4, 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 = [11, 4, 4]
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
source roles = [11, 4]
After
derived = [[[1, 1], [2, 4], [3, 9], [4, 5], [5, 3], [6, 3], [7, 5], [8, 9], [9, 4], [10, 1]], [2, 9], 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, S016
Before
Euler binary power accumulate0 := mul(1, 4)
After
Euler binary power accumulate0 = 4
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 binary power square0 := mul(4, 4)
After
Euler binary power square0 = 16
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
source roles = [0, [1, 4, 5], 11, 4, 5]
After
derived = [4, 16, 4, 5, 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, S019
Before
Euler binary power square1 := mul(5, 5)
After
Euler binary power square1 = 25
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
source roles = [1, [4, 5, 2], 11, 4, 5]
After
derived = [null, 25, 4, 3, 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, S021
Before
Euler binary power accumulate2 := mul(4, 3)
After
Euler binary power accumulate2 = 12
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 binary power square2 := mul(3, 3)
After
Euler binary power square2 = 9
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
source roles = [2, [4, 3, 1], 11, 4, 5]
After
derived = [12, 9, 1, 9, 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, S024
Before
source roles = [11, [2, 9], 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, S025
S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Recompute Euler binary power square2 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute Euler binary power accumulate2 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Recompute Euler binary power square1 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute Euler binary power square0 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute Euler binary power accumulate0 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(11, 4, [2, 9], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations)
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: b74da33ad5d0dd2fdfc80d9d8b472d86061be75612b0a78ce4c0ff7f6e587cb4candidate record SHA-256: 709be37a9111f1ed739366540216f6fbb2b1dda38623d2100c39a3717747794asemantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED