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R0047-24177 · q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0052
Use odd prime p=5, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).
G001 — Use odd prime p=5, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, 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-residue-nonresidue-cases]
After
derived = [[prime, normalized_class, classification, nonzero_roots, Legendre_symbol, 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 = [5]
After
derived = [[[2, 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, 5)
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(2, 0)
After
Euler ORIGINAL signed a minus modulus multiple = 2
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 = [2, 5, 0]
After
derived = [2, 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 = [5, 2, 2]
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
source roles = [5, 2]
After
derived = [[[1, 1], [2, 4], [3, 4], [4, 1]], [], -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, S010
Before
Euler binary power square0 := mul(2, 2)
After
Euler binary power square0 = 4
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
source roles = [0, [1, 2, 2], 5, 2, 2]
After
derived = [null, 4, 1, 4, 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, S012
Before
Euler binary power accumulate1 := mul(1, 4)
After
Euler binary power accumulate1 = 4
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 binary power square1 := mul(4, 4)
After
Euler binary power square1 = 16
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
source roles = [1, [1, 4, 1], 5, 2, 2]
After
derived = [4, 16, 4, 1, 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, S015
Before
source roles = [5, [], -1, 4]
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, S016
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Recompute Euler binary power square1 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute Euler binary power accumulate1 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute Euler binary power square0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(5, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)
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: 500f2e399b8dbbd6c5dc97b2326e4a2d615e2b5b33be1411cfbae831ef07c4e3candidate record SHA-256: 1ac9181944de02f7fe66ff92d64abfc96b3698fc5aa60ccf4a7b5fb835aa66b9semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED