Question 19832: A Composite Can Pass a Fermat Test for a Chosen Base

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0038-19832 · q-20261014-advance-20261106-r05-fermat-little-theorem-0082

Original question and givens

Use integer base a=13 and modulus=561, normalized residues in [0,modulus). Exact integer square-and-multiply computes powers; trial division checks primality. Every equality of residues here is a congruence modulo the stated modulus, not an unrestricted integer equality. The coprime form requires prime modulus and an invertible base; the all-integer form for prime p also allows p to divide a. Never cancel a noninvertible residue. Give (modulus, base, prime, coprime, power_modulus_minus_1, boundary).

G001 — Use integer base a=13 and modulus=561, normalized residues in [0,modulus). Exact integer square-and-multiply computes powers; trial division checks primality. Every equality of residues here is a congruence modulo the stated modulus, not an unrestricted integer equality. The coprime form requires prime modulus and an invertible base; the all-integer form for prime p also allows p to divide a. Never cancel a noninvertible residue. Give (modulus, base, prime, coprime, power_modulus_minus_1, boundary).

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-fermat-little-theorem

Step-by-step derivation

S001 · implies

Before

source operation=flt-converse-counterexample;original snapshot and mathematics retained

After

explicit ordered source output fields=[modulus, base, prime, coprime, power_modulus_minus_1, boundary]

Conditions: none

Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ACTUAL integer modulus=561;complete trial range2..floor_sqrt=23

After

checked rows[divisor,quotient,remainder]=[[2, 280, 1], [3, 187, 0]];first exact factor=3;prime=No

Conditions: none

An exact factor in the trial range witnesses compositeness.If the entire range has no factor,any composite decomposition would require one at most sqrt(n),so all composite cases are excluded.The empty range for2 is valid.This is not a primality inference from a Fermat test.

Step identifiers and dependencies

Rule: flt_primality_complete_trial_or_actual_factor_not_Fermat_test · Uses: G001, S001

S003 · implies

Before

original signed base=13;positive modulus=561;use nonnegative gcd inputs=[13, 561]

After

ALL Euclidean rows[left,right,q,r]=[[13, 561, 0, 13], [561, 13, 43, 2], [13, 2, 6, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=Yes

Conditions: none

Euclidean replacements preserve common divisors and give the terminal gcd,including gcd(0,m)=m.Primality of the modulus and invertibility of the base remain separate flags.

Step identifiers and dependencies

Rule: flt_actual_gcd_coprime_separate_from_primality · Uses: G001, S002

S004 · calculation

Before

Fermat original signed base exact quotient := div(13, 561)

After

Fermat original signed base exact quotient = 13/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Fermat original signed base floor quotient multiple := mul(0, 561)

After

Fermat original signed base floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Fermat original signed base normalized remainder := sub(13, 0)

After

Fermat original signed base normalized remainder = 13

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

Fermat original signed base:signed integer value=13;positive modulus=561;exact quotient=13/561;floor q=0

After

value=q*modulus+remainder;normalized remainder=13 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S006

S008 · implies

Before

Fermat original modular exponent:original normalized base=13;original exponent=560;modulus=561

After

initialize accumulator1,current_base=base,remaining=exponent;invariant accumulator*current_base^remaining congruent to original power

Conditions: none

The multiplicative identity gives the exponent0 convention.Each step preserves congruence to the same original power without constructing huge ordinary powers or replacing modular relations with real equalities.

Step identifiers and dependencies

Rule: flt_modular_power_initial_invariant · Uses: G001, S007

S009 · calculation

Before

Fermat original modular exponent bit0 current base square := mul(13, 13)

After

Fermat original modular exponent bit0 current base square = 169

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Fermat original modular exponent bit0 squared base reduction exact quotient := div(169, 561)

After

Fermat original modular exponent bit0 squared base reduction exact quotient = 169/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(0, 561)

After

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(169, 0)

After

Fermat original modular exponent bit0 squared base reduction normalized remainder = 169

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

Fermat original modular exponent bit0 squared base reduction:signed integer value=169;positive modulus=561;exact quotient=169/561;floor q=0

After

value=q*modulus+remainder;normalized remainder=169 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S012

S014 · implies

Before

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 13, 560];low bit=0

After

next[accumulator,base,exponent]=[1, 169, 280];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Replace an even power by a squared base;an odd power also contributes one base factor to the accumulator.Exact reductions preserve congruence and halving the remaining nonnegative exponent guarantees termination.

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S013

S015 · calculation

Before

Fermat original modular exponent bit1 current base square := mul(169, 169)

After

Fermat original modular exponent bit1 current base square = 28561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Fermat original modular exponent bit1 squared base reduction exact quotient := div(28561, 561)

After

Fermat original modular exponent bit1 squared base reduction exact quotient = 28561/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(50, 561)

After

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 28050

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(28561, 28050)

After

Fermat original modular exponent bit1 squared base reduction normalized remainder = 511

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

Fermat original modular exponent bit1 squared base reduction:signed integer value=28561;positive modulus=561;exact quotient=28561/561;floor q=50

After

value=q*modulus+remainder;normalized remainder=511 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S018

S020 · implies

Before

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 169, 280];low bit=0

After

next[accumulator,base,exponent]=[1, 511, 140];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S019

S021 · calculation

Before

Fermat original modular exponent bit2 current base square := mul(511, 511)

After

Fermat original modular exponent bit2 current base square = 261121

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Fermat original modular exponent bit2 squared base reduction exact quotient := div(261121, 561)

After

Fermat original modular exponent bit2 squared base reduction exact quotient = 261121/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(465, 561)

After

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 260865

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(261121, 260865)

After

Fermat original modular exponent bit2 squared base reduction normalized remainder = 256

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

Fermat original modular exponent bit2 squared base reduction:signed integer value=261121;positive modulus=561;exact quotient=261121/561;floor q=465

After

value=q*modulus+remainder;normalized remainder=256 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S024

S026 · implies

Before

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 511, 140];low bit=0

After

next[accumulator,base,exponent]=[1, 256, 70];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S025

S027 · calculation

Before

Fermat original modular exponent bit3 current base square := mul(256, 256)

After

Fermat original modular exponent bit3 current base square = 65536

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Fermat original modular exponent bit3 squared base reduction exact quotient := div(65536, 561)

After

Fermat original modular exponent bit3 squared base reduction exact quotient = 65536/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(116, 561)

After

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 65076

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(65536, 65076)

After

Fermat original modular exponent bit3 squared base reduction normalized remainder = 460

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

Fermat original modular exponent bit3 squared base reduction:signed integer value=65536;positive modulus=561;exact quotient=65536/561;floor q=116

After

value=q*modulus+remainder;normalized remainder=460 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S030

S032 · implies

Before

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 256, 70];low bit=0

After

next[accumulator,base,exponent]=[1, 460, 35];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S031

S033 · calculation

Before

Fermat original modular exponent bit4 accumulator product := mul(1, 460)

After

Fermat original modular exponent bit4 accumulator product = 460

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(460, 561)

After

Fermat original modular exponent bit4 accumulator reduction exact quotient = 460/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(0, 561)

After

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(460, 0)

After

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 460

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · implies

Before

Fermat original modular exponent bit4 accumulator reduction:signed integer value=460;positive modulus=561;exact quotient=460/561;floor q=0

After

value=q*modulus+remainder;normalized remainder=460 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S036

S038 · calculation

Before

Fermat original modular exponent bit4 current base square := mul(460, 460)

After

Fermat original modular exponent bit4 current base square = 211600

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Fermat original modular exponent bit4 squared base reduction exact quotient := div(211600, 561)

After

Fermat original modular exponent bit4 squared base reduction exact quotient = 211600/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(377, 561)

After

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 211497

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(211600, 211497)

After

Fermat original modular exponent bit4 squared base reduction normalized remainder = 103

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

Fermat original modular exponent bit4 squared base reduction:signed integer value=211600;positive modulus=561;exact quotient=211600/561;floor q=377

After

value=q*modulus+remainder;normalized remainder=103 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S041

S043 · implies

Before

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[1, 460, 35];low bit=1

After

next[accumulator,base,exponent]=[460, 103, 17];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S042

S044 · calculation

Before

Fermat original modular exponent bit5 accumulator product := mul(460, 103)

After

Fermat original modular exponent bit5 accumulator product = 47380

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(47380, 561)

After

Fermat original modular exponent bit5 accumulator reduction exact quotient = 47380/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(84, 561)

After

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 47124

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(47380, 47124)

After

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 256

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

Fermat original modular exponent bit5 accumulator reduction:signed integer value=47380;positive modulus=561;exact quotient=47380/561;floor q=84

After

value=q*modulus+remainder;normalized remainder=256 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S047

S049 · calculation

Before

Fermat original modular exponent bit5 current base square := mul(103, 103)

After

Fermat original modular exponent bit5 current base square = 10609

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

Fermat original modular exponent bit5 squared base reduction exact quotient := div(10609, 561)

After

Fermat original modular exponent bit5 squared base reduction exact quotient = 10609/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(18, 561)

After

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 10098

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(10609, 10098)

After

Fermat original modular exponent bit5 squared base reduction normalized remainder = 511

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · implies

Before

Fermat original modular exponent bit5 squared base reduction:signed integer value=10609;positive modulus=561;exact quotient=10609/561;floor q=18

After

value=q*modulus+remainder;normalized remainder=511 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S052

S054 · implies

Before

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[460, 103, 17];low bit=1

After

next[accumulator,base,exponent]=[256, 511, 8];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S053

S055 · calculation

Before

Fermat original modular exponent bit6 current base square := mul(511, 511)

After

Fermat original modular exponent bit6 current base square = 261121

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

Fermat original modular exponent bit6 squared base reduction exact quotient := div(261121, 561)

After

Fermat original modular exponent bit6 squared base reduction exact quotient = 261121/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(465, 561)

After

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 260865

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(261121, 260865)

After

Fermat original modular exponent bit6 squared base reduction normalized remainder = 256

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

Fermat original modular exponent bit6 squared base reduction:signed integer value=261121;positive modulus=561;exact quotient=261121/561;floor q=465

After

value=q*modulus+remainder;normalized remainder=256 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S058

S060 · implies

Before

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[256, 511, 8];low bit=0

After

next[accumulator,base,exponent]=[256, 256, 4];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S059

S061 · calculation

Before

Fermat original modular exponent bit7 current base square := mul(256, 256)

After

Fermat original modular exponent bit7 current base square = 65536

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

Fermat original modular exponent bit7 squared base reduction exact quotient := div(65536, 561)

After

Fermat original modular exponent bit7 squared base reduction exact quotient = 65536/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

Fermat original modular exponent bit7 squared base reduction floor quotient multiple := mul(116, 561)

After

Fermat original modular exponent bit7 squared base reduction floor quotient multiple = 65076

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

Fermat original modular exponent bit7 squared base reduction normalized remainder := sub(65536, 65076)

After

Fermat original modular exponent bit7 squared base reduction normalized remainder = 460

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · implies

Before

Fermat original modular exponent bit7 squared base reduction:signed integer value=65536;positive modulus=561;exact quotient=65536/561;floor q=116

After

value=q*modulus+remainder;normalized remainder=460 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S064

S066 · implies

Before

Fermat original modular exponent:step=7;previous[accumulator,base,exponent]=[256, 256, 4];low bit=0

After

next[accumulator,base,exponent]=[256, 460, 2];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S065

S067 · calculation

Before

Fermat original modular exponent bit8 current base square := mul(460, 460)

After

Fermat original modular exponent bit8 current base square = 211600

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Fermat original modular exponent bit8 squared base reduction exact quotient := div(211600, 561)

After

Fermat original modular exponent bit8 squared base reduction exact quotient = 211600/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Fermat original modular exponent bit8 squared base reduction floor quotient multiple := mul(377, 561)

After

Fermat original modular exponent bit8 squared base reduction floor quotient multiple = 211497

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

Fermat original modular exponent bit8 squared base reduction normalized remainder := sub(211600, 211497)

After

Fermat original modular exponent bit8 squared base reduction normalized remainder = 103

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · implies

Before

Fermat original modular exponent bit8 squared base reduction:signed integer value=211600;positive modulus=561;exact quotient=211600/561;floor q=377

After

value=q*modulus+remainder;normalized remainder=103 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S070

S072 · implies

Before

Fermat original modular exponent:step=8;previous[accumulator,base,exponent]=[256, 460, 2];low bit=0

After

next[accumulator,base,exponent]=[256, 103, 1];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S071

S073 · calculation

Before

Fermat original modular exponent bit9 accumulator product := mul(256, 103)

After

Fermat original modular exponent bit9 accumulator product = 26368

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

Fermat original modular exponent bit9 accumulator reduction exact quotient := div(26368, 561)

After

Fermat original modular exponent bit9 accumulator reduction exact quotient = 26368/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

Fermat original modular exponent bit9 accumulator reduction floor quotient multiple := mul(47, 561)

After

Fermat original modular exponent bit9 accumulator reduction floor quotient multiple = 26367

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

Fermat original modular exponent bit9 accumulator reduction normalized remainder := sub(26368, 26367)

After

Fermat original modular exponent bit9 accumulator reduction normalized remainder = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · implies

Before

Fermat original modular exponent bit9 accumulator reduction:signed integer value=26368;positive modulus=561;exact quotient=26368/561;floor q=47

After

value=q*modulus+remainder;normalized remainder=1 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S076

S078 · calculation

Before

Fermat original modular exponent bit9 current base square := mul(103, 103)

After

Fermat original modular exponent bit9 current base square = 10609

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

Fermat original modular exponent bit9 squared base reduction exact quotient := div(10609, 561)

After

Fermat original modular exponent bit9 squared base reduction exact quotient = 10609/561

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

Before

Fermat original modular exponent bit9 squared base reduction floor quotient multiple := mul(18, 561)

After

Fermat original modular exponent bit9 squared base reduction floor quotient multiple = 10098

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · calculation

Before

Fermat original modular exponent bit9 squared base reduction normalized remainder := sub(10609, 10098)

After

Fermat original modular exponent bit9 squared base reduction normalized remainder = 511

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · implies

Before

Fermat original modular exponent bit9 squared base reduction:signed integer value=10609;positive modulus=561;exact quotient=10609/561;floor q=18

After

value=q*modulus+remainder;normalized remainder=511 in[0,561)

Conditions: none

Use a positive modulus and floor quotient to obtain the nonnegative representative even for a negative base.Equal remainders express congruence modulo this modulus,not unrestricted integer equality.

Step identifiers and dependencies

Rule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S081

S083 · implies

Before

Fermat original modular exponent:step=9;previous[accumulator,base,exponent]=[256, 103, 1];low bit=1

After

next[accumulator,base,exponent]=[1, 511, 0];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: flt_each_binary_power_step_exact_invariant · Uses: G001, S082

S084 · implies

Before

Fermat original modular exponent:remaining exponent0;final normalized accumulator=1

After

EXACT original base^560 modulo561=1

Conditions: none

At remaining exponent0 the invariant leaves exactly the accumulator,giving the original power's residue.The arithmetic result alone does not certify a prime modulus.

Step identifiers and dependencies

Rule: flt_modular_power_terminal_not_primality_converse · Uses: G001, S083

S085 · implies

Before

ACTUAL composite modulus=561 certified by a trial factor;ORIGINAL base=13 coprime;computed a^(m-1) modulo m=1

After

THIS chosen composite/base pair passes the Fermat power test,but modulus is NOT prime;counterexample to converse ONLY

Conditions: none

This explicit composite/coprime-base pair refutes the implication from a passing test to primality.It says neither that every composite nor every base passes,and does not refute Fermat's theorem under its prime premise.

Step identifiers and dependencies

Rule: flt_composite_passing_chosen_base_not_all_bases_or_theorem_failure · Uses: G001, S084

Reverse checks

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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S081 · premise_and_result_audit — Recompute Fermat original modular exponent bit9 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S080 · premise_and_result_audit — Recompute Fermat original modular exponent bit9 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S079 · premise_and_result_audit — Recompute Fermat original modular exponent bit9 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S078 · premise_and_result_audit — Recompute Fermat original modular exponent bit9 current base square 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 Fermat original modular exponent bit9 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S075 · premise_and_result_audit — Recompute Fermat original modular exponent bit9 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S074 · premise_and_result_audit — Recompute Fermat original modular exponent bit9 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S073 · premise_and_result_audit — Recompute Fermat original modular exponent bit9 accumulator product with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S070 · premise_and_result_audit — Recompute Fermat original modular exponent bit8 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S069 · premise_and_result_audit — Recompute Fermat original modular exponent bit8 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S068 · premise_and_result_audit — Recompute Fermat original modular exponent bit8 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S067 · premise_and_result_audit — Recompute Fermat original modular exponent bit8 current base square with exact rational arithmetic, not rounded decimals.
S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S064 · premise_and_result_audit — Recompute Fermat original modular exponent bit7 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S063 · premise_and_result_audit — Recompute Fermat original modular exponent bit7 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S062 · premise_and_result_audit — Recompute Fermat original modular exponent bit7 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S061 · premise_and_result_audit — Recompute Fermat original modular exponent bit7 current base square with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S058 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S057 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 current base square with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S052 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 current base square 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 Fermat original modular exponent bit5 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 accumulator product 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S041 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 current base square with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S036 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator product with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S030 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 current base square with exact rational arithmetic, not rounded decimals.
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 Fermat original modular exponent bit2 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 current base square 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 current base square with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 Fermat original modular exponent bit0 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 current base square with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Recompute Fermat original signed base normalized remainder with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Fermat original signed base floor quotient multiple with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Fermat original signed base exact quotient with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

(561, 13, No, Yes, 1, converse_false_composite_can_pass_Fermat_test)

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: 69c97a8c5d94c34155acdbd6131637d339e39a548525c84e98c34562a583c370
candidate record SHA-256: 178d9b3a2f1fd1fbec85f4e66c59d9a0a92e388aa52da6c0934491091bed7776
semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON