# Question 24169: Compare a Square-Table Legendre Value with a Separately Computed Half Power

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0047-24169 · q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0044



## Original question and givens

Use odd prime p=71, original a=19. Give (prime, normalized_class, nonzero_roots, Legendre_symbol, modular_power, signed_power, equal, certificate).


- G001 — Use odd prime p=71, original a=19. Give (prime, normalized_class, nonzero_roots, Legendre_symbol, modular_power, signed_power, equal, certificate).





## Definition and scope

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.


- [mko-euler-criterion-quadratic-residue](https://ocme.evemisslab.com/en/mko/mko-euler-criterion-quadratic-residue/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [71]

After

derived = [[[2, 1], [3, 2], [4, 3], [5, 1], [6, 5], [7, 1], [8, 7]], 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.

Step identifiers and dependenciesRule: Euler_modulus_primality_complete_trial · Uses: G001, S001





S003 · calculationBefore

Euler ORIGINAL signed quotient times modulus := mul(0, 71)

After

Euler ORIGINAL signed quotient times modulus = 0

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

Euler ORIGINAL signed a minus modulus multiple := sub(19, 0)

After

Euler ORIGINAL signed a minus modulus multiple = 19

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

source roles = [19, 71, 0]

After

derived = [19, Yes]

Conditions: none

Use a=qp+r with0<=r<p,including negative a;retain original a where the interface requests it.

Step identifiers and dependenciesRule: Euler_Euclidean_class_before_hypothesis · Uses: G001, S004





S006 · impliesBefore

source roles = [71, 19, 19]

After

derived = [Yes, Yes]

Conditions: none

Qualify odd primality and the nonzero class first.Define Legendre by the square table before comparing a power.

Step identifiers and dependenciesRule: Euler_odd_prime_nonzero_qualified · Uses: G001, S005





S007 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033





S035 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037





S039 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041





S043 · calculationBefore

Euler square root candidate37 := mul(37, 37)

After

Euler square root candidate37 = 1369

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · calculationBefore

Euler square root candidate38 := mul(38, 38)

After

Euler square root candidate38 = 1444

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · calculationBefore

Euler square root candidate39 := mul(39, 39)

After

Euler square root candidate39 = 1521

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · calculationBefore

Euler square root candidate40 := mul(40, 40)

After

Euler square root candidate40 = 1600

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045





S047 · calculationBefore

Euler square root candidate41 := mul(41, 41)

After

Euler square root candidate41 = 1681

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · calculationBefore

Euler square root candidate42 := mul(42, 42)

After

Euler square root candidate42 = 1764

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · calculationBefore

Euler square root candidate43 := mul(43, 43)

After

Euler square root candidate43 = 1849

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · calculationBefore

Euler square root candidate44 := mul(44, 44)

After

Euler square root candidate44 = 1936

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049





S051 · calculationBefore

Euler square root candidate45 := mul(45, 45)

After

Euler square root candidate45 = 2025

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050





S052 · calculationBefore

Euler square root candidate46 := mul(46, 46)

After

Euler square root candidate46 = 2116

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051





S053 · calculationBefore

Euler square root candidate47 := mul(47, 47)

After

Euler square root candidate47 = 2209

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · calculationBefore

Euler square root candidate48 := mul(48, 48)

After

Euler square root candidate48 = 2304

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053





S055 · calculationBefore

Euler square root candidate49 := mul(49, 49)

After

Euler square root candidate49 = 2401

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · calculationBefore

Euler square root candidate50 := mul(50, 50)

After

Euler square root candidate50 = 2500

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055





S057 · calculationBefore

Euler square root candidate51 := mul(51, 51)

After

Euler square root candidate51 = 2601

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056





S058 · calculationBefore

Euler square root candidate52 := mul(52, 52)

After

Euler square root candidate52 = 2704

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057





S059 · calculationBefore

Euler square root candidate53 := mul(53, 53)

After

Euler square root candidate53 = 2809

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058





S060 · calculationBefore

Euler square root candidate54 := mul(54, 54)

After

Euler square root candidate54 = 2916

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059





S061 · calculationBefore

Euler square root candidate55 := mul(55, 55)

After

Euler square root candidate55 = 3025

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · calculationBefore

Euler square root candidate56 := mul(56, 56)

After

Euler square root candidate56 = 3136

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061





S063 · calculationBefore

Euler square root candidate57 := mul(57, 57)

After

Euler square root candidate57 = 3249

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062





S064 · calculationBefore

Euler square root candidate58 := mul(58, 58)

After

Euler square root candidate58 = 3364

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063





S065 · calculationBefore

Euler square root candidate59 := mul(59, 59)

After

Euler square root candidate59 = 3481

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064





S066 · calculationBefore

Euler square root candidate60 := mul(60, 60)

After

Euler square root candidate60 = 3600

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065





S067 · calculationBefore

Euler square root candidate61 := mul(61, 61)

After

Euler square root candidate61 = 3721

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066





S068 · calculationBefore

Euler square root candidate62 := mul(62, 62)

After

Euler square root candidate62 = 3844

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067





S069 · calculationBefore

Euler square root candidate63 := mul(63, 63)

After

Euler square root candidate63 = 3969

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068





S070 · calculationBefore

Euler square root candidate64 := mul(64, 64)

After

Euler square root candidate64 = 4096

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069





S071 · calculationBefore

Euler square root candidate65 := mul(65, 65)

After

Euler square root candidate65 = 4225

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070





S072 · calculationBefore

Euler square root candidate66 := mul(66, 66)

After

Euler square root candidate66 = 4356

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071





S073 · calculationBefore

Euler square root candidate67 := mul(67, 67)

After

Euler square root candidate67 = 4489

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072





S074 · calculationBefore

Euler square root candidate68 := mul(68, 68)

After

Euler square root candidate68 = 4624

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073





S075 · calculationBefore

Euler square root candidate69 := mul(69, 69)

After

Euler square root candidate69 = 4761

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074





S076 · calculationBefore

Euler square root candidate70 := mul(70, 70)

After

Euler square root candidate70 = 4900

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075





S077 · impliesBefore

source roles = [71, 19]

After

derived = [[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 10], [10, 29], [11, 50], [12, 2], [13, 27], [14, 54], [15, 12], [16, 43], [17, 5], [18, 40], [19, 6], [20, 45], [21, 15], [22, 58], [23, 32], [24, 8], [25, 57], [26, 37], [27, 19], [28, 3], [29, 60], [30, 48], [31, 38], [32, 30], [33, 24], [34, 20], [35, 18], [36, 18], [37, 20], [38, 24], [39, 30], [40, 38], [41, 48], [42, 60], [43, 3], [44, 19], [45, 37], [46, 57], [47, 8], [48, 32], [49, 58], [50, 15], [51, 45], [52, 6], [53, 40], [54, 5], [55, 43], [56, 12], [57, 54], [58, 27], [59, 2], [60, 50], [61, 29], [62, 10], [63, 64], [64, 49], [65, 36], [66, 25], [67, 16], [68, 9], [69, 4], [70, 1]], [27, 44], 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.

Step identifiers and dependenciesRule: Euler_ALL_nonzero_square_table_defines_Legendre · Uses: G001, S076





S078 · calculationBefore

Euler binary power accumulate0 := mul(1, 19)

After

Euler binary power accumulate0 = 19

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077





S079 · calculationBefore

Euler binary power square0 := mul(19, 19)

After

Euler binary power square0 = 361

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078





S080 · impliesBefore

source roles = [0, [1, 19, 35], 71, 19, 35]

After

derived = [19, 361, 19, 6, 17]

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.

Step identifiers and dependenciesRule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S079





S081 · calculationBefore

Euler binary power accumulate1 := mul(19, 6)

After

Euler binary power accumulate1 = 114

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080





S082 · calculationBefore

Euler binary power square1 := mul(6, 6)

After

Euler binary power square1 = 36

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081





S083 · impliesBefore

source roles = [1, [19, 6, 17], 71, 19, 35]

After

derived = [114, 36, 43, 36, 8]

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.

Step identifiers and dependenciesRule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S082





S084 · calculationBefore

Euler binary power square2 := mul(36, 36)

After

Euler binary power square2 = 1296

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083





S085 · impliesBefore

source roles = [2, [43, 36, 8], 71, 19, 35]

After

derived = [null, 1296, 43, 18, 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.

Step identifiers and dependenciesRule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S084





S086 · calculationBefore

Euler binary power square3 := mul(18, 18)

After

Euler binary power square3 = 324

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085





S087 · impliesBefore

source roles = [3, [43, 18, 4], 71, 19, 35]

After

derived = [null, 324, 43, 40, 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.

Step identifiers and dependenciesRule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S086





S088 · calculationBefore

Euler binary power square4 := mul(40, 40)

After

Euler binary power square4 = 1600

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087





S089 · impliesBefore

source roles = [4, [43, 40, 2], 71, 19, 35]

After

derived = [null, 1600, 43, 38, 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.

Step identifiers and dependenciesRule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S088





S090 · calculationBefore

Euler binary power accumulate5 := mul(43, 38)

After

Euler binary power accumulate5 = 1634

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089





S091 · calculationBefore

Euler binary power square5 := mul(38, 38)

After

Euler binary power square5 = 1444

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090





S092 · impliesBefore

source roles = [5, [43, 38, 1], 71, 19, 35]

After

derived = [1634, 1444, 1, 24, 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.

Step identifiers and dependenciesRule: Euler_repeated_squaring_original_exponent_invariant · Uses: G001, S091





S093 · impliesBefore

source roles = [71, [27, 44], 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.

Step identifiers and dependenciesRule: Euler_decode_p_minus1_and_instance_only_comparison · Uses: G001, S092









## Reverse checks


- S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S091 · premise_and_result_audit — Recompute Euler binary power square5 with exact rational arithmetic, not rounded decimals.
- S090 · premise_and_result_audit — Recompute Euler binary power accumulate5 with exact rational arithmetic, not rounded decimals.
- S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S088 · premise_and_result_audit — Recompute Euler binary power square4 with exact rational arithmetic, not rounded decimals.
- S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S086 · premise_and_result_audit — Recompute Euler binary power square3 with exact rational arithmetic, not rounded decimals.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Recompute Euler binary power square2 with exact rational arithmetic, not rounded decimals.
- S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S082 · premise_and_result_audit — Recompute Euler binary power square1 with exact rational arithmetic, not rounded decimals.
- S081 · premise_and_result_audit — Recompute Euler binary power accumulate1 with exact rational arithmetic, not rounded decimals.
- S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S079 · premise_and_result_audit — Recompute Euler binary power square0 with exact rational arithmetic, not rounded decimals.
- S078 · premise_and_result_audit — Recompute Euler binary power accumulate0 with exact rational arithmetic, not rounded decimals.
- S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S076 · premise_and_result_audit — Recompute Euler square root candidate70 with exact rational arithmetic, not rounded decimals.
- S075 · premise_and_result_audit — Recompute Euler square root candidate69 with exact rational arithmetic, not rounded decimals.
- S074 · premise_and_result_audit — Recompute Euler square root candidate68 with exact rational arithmetic, not rounded decimals.
- S073 · premise_and_result_audit — Recompute Euler square root candidate67 with exact rational arithmetic, not rounded decimals.
- S072 · premise_and_result_audit — Recompute Euler square root candidate66 with exact rational arithmetic, not rounded decimals.
- S071 · premise_and_result_audit — Recompute Euler square root candidate65 with exact rational arithmetic, not rounded decimals.
- S070 · premise_and_result_audit — Recompute Euler square root candidate64 with exact rational arithmetic, not rounded decimals.
- S069 · premise_and_result_audit — Recompute Euler square root candidate63 with exact rational arithmetic, not rounded decimals.
- S068 · premise_and_result_audit — Recompute Euler square root candidate62 with exact rational arithmetic, not rounded decimals.
- S067 · premise_and_result_audit — Recompute Euler square root candidate61 with exact rational arithmetic, not rounded decimals.
- S066 · premise_and_result_audit — Recompute Euler square root candidate60 with exact rational arithmetic, not rounded decimals.
- S065 · premise_and_result_audit — Recompute Euler square root candidate59 with exact rational arithmetic, not rounded decimals.
- S064 · premise_and_result_audit — Recompute Euler square root candidate58 with exact rational arithmetic, not rounded decimals.
- S063 · premise_and_result_audit — Recompute Euler square root candidate57 with exact rational arithmetic, not rounded decimals.
- S062 · premise_and_result_audit — Recompute Euler square root candidate56 with exact rational arithmetic, not rounded decimals.
- S061 · premise_and_result_audit — Recompute Euler square root candidate55 with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Recompute Euler square root candidate54 with exact rational arithmetic, not rounded decimals.
- S059 · premise_and_result_audit — Recompute Euler square root candidate53 with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Recompute Euler square root candidate52 with exact rational arithmetic, not rounded decimals.
- S057 · premise_and_result_audit — Recompute Euler square root candidate51 with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Recompute Euler square root candidate50 with exact rational arithmetic, not rounded decimals.
- S055 · premise_and_result_audit — Recompute Euler square root candidate49 with exact rational arithmetic, not rounded decimals.
- S054 · premise_and_result_audit — Recompute Euler square root candidate48 with exact rational arithmetic, not rounded decimals.
- S053 · premise_and_result_audit — Recompute Euler square root candidate47 with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Recompute Euler square root candidate46 with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Recompute Euler square root candidate45 with exact rational arithmetic, not rounded decimals.
- S050 · premise_and_result_audit — Recompute Euler square root candidate44 with exact rational arithmetic, not rounded decimals.
- S049 · premise_and_result_audit — Recompute Euler square root candidate43 with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute Euler square root candidate42 with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Recompute Euler square root candidate41 with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Recompute Euler square root candidate40 with exact rational arithmetic, not rounded decimals.
- S045 · premise_and_result_audit — Recompute Euler square root candidate39 with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Recompute Euler square root candidate38 with exact rational arithmetic, not rounded decimals.
- S043 · premise_and_result_audit — Recompute Euler square root candidate37 with exact rational arithmetic, not rounded decimals.
- 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.





## Result

(71, 19, [27, 44], 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.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 90681c1d8779e94d7ec7c0c48e5f2c1808b550708e8ed49a452f67aace1e3e9c
- candidate record SHA-256: 90317b03683853b232b34e75ed39111db1706c3a9886da6fd4e1814803cd0473
- semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386
- question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0044.json)