Question 19817: The All-Integer Prime Form Includes Bases Divisible by p

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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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R0038-19817 · q-20261014-advance-20261106-r05-fermat-little-theorem-0042

Original question and givens

Use integer base a=1003 and modulus=59, 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 (p, normalized_base, power_p, comparison_residue, congruent, boundary).

G001 — Use integer base a=1003 and modulus=59, 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 (p, normalized_base, power_p, comparison_residue, congruent, 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-all-integer;original snapshot and mathematics retained

After

explicit ordered source output fields=[p, normalized_base, power_p, comparison_residue, congruent, 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=59;complete trial range2..floor_sqrt=7

After

checked rows[divisor,quotient,remainder]=[[2, 29, 1], [3, 19, 2], [4, 14, 3], [5, 11, 4], [6, 9, 5], [7, 8, 3]];first exact factor=null;prime=Yes

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=1003;positive modulus=59;use nonnegative gcd inputs=[1003, 59]

After

ALL Euclidean rows[left,right,q,r]=[[1003, 59, 17, 0]];terminal nonnegative gcd=59;coprime=No

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(1003, 59)

After

Fermat original signed base exact quotient = 17

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(17, 59)

After

Fermat original signed base floor quotient multiple = 1003

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(1003, 1003)

After

Fermat original signed base normalized remainder = 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, S005

S007 · implies

Before

Fermat original signed base:signed integer value=1003;positive modulus=59;exact quotient=17;floor q=17

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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=0;original exponent=59;modulus=59

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 accumulator product := mul(1, 0)

After

Fermat original modular exponent bit0 accumulator product = 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, S008

S010 · calculation

Before

Fermat original modular exponent bit0 accumulator reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit0 accumulator reduction exact quotient = 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, S009

S011 · calculation

Before

Fermat original modular exponent bit0 accumulator reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit0 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, S010

S012 · calculation

Before

Fermat original modular exponent bit0 accumulator reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit0 accumulator reduction normalized remainder = 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, S011

S013 · implies

Before

Fermat original modular exponent bit0 accumulator reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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 · calculation

Before

Fermat original modular exponent bit0 current base square := mul(0, 0)

After

Fermat original modular exponent bit0 current base square = 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, S013

S015 · calculation

Before

Fermat original modular exponent bit0 squared base reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit0 squared base reduction exact quotient = 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, S014

S016 · calculation

Before

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(0, 59)

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, S015

S017 · calculation

Before

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

After

Fermat original modular exponent bit0 squared base reduction normalized remainder = 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, S016

S018 · implies

Before

Fermat original modular exponent bit0 squared base reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S017

S019 · implies

Before

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 0, 59];low bit=1

After

next[accumulator,base,exponent]=[0, 0, 29];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, S018

S020 · calculation

Before

Fermat original modular exponent bit1 accumulator product := mul(0, 0)

After

Fermat original modular exponent bit1 accumulator product = 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, S019

S021 · calculation

Before

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit1 accumulator reduction exact quotient = 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, S020

S022 · calculation

Before

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit1 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, S021

S023 · calculation

Before

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 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, S022

S024 · implies

Before

Fermat original modular exponent bit1 accumulator reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S023

S025 · calculation

Before

Fermat original modular exponent bit1 current base square := mul(0, 0)

After

Fermat original modular exponent bit1 current base square = 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, S024

S026 · calculation

Before

Fermat original modular exponent bit1 squared base reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit1 squared base reduction exact quotient = 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, S025

S027 · calculation

Before

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit1 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, S026

S028 · calculation

Before

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit1 squared base reduction normalized remainder = 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, S027

S029 · implies

Before

Fermat original modular exponent bit1 squared base reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S028

S030 · implies

Before

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[0, 0, 29];low bit=1

After

next[accumulator,base,exponent]=[0, 0, 14];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, S029

S031 · calculation

Before

Fermat original modular exponent bit2 current base square := mul(0, 0)

After

Fermat original modular exponent bit2 current base square = 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, S030

S032 · calculation

Before

Fermat original modular exponent bit2 squared base reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit2 squared base reduction exact quotient = 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, S031

S033 · calculation

Before

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit2 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, S032

S034 · calculation

Before

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit2 squared base reduction normalized remainder = 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, S033

S035 · implies

Before

Fermat original modular exponent bit2 squared base reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S034

S036 · implies

Before

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[0, 0, 14];low bit=0

After

next[accumulator,base,exponent]=[0, 0, 7];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, S035

S037 · calculation

Before

Fermat original modular exponent bit3 accumulator product := mul(0, 0)

After

Fermat original modular exponent bit3 accumulator product = 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, S036

S038 · calculation

Before

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit3 accumulator reduction exact quotient = 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, S037

S039 · calculation

Before

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit3 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, S038

S040 · calculation

Before

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 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, S039

S041 · implies

Before

Fermat original modular exponent bit3 accumulator reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S040

S042 · calculation

Before

Fermat original modular exponent bit3 current base square := mul(0, 0)

After

Fermat original modular exponent bit3 current base square = 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, S041

S043 · calculation

Before

Fermat original modular exponent bit3 squared base reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit3 squared base reduction exact quotient = 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, S042

S044 · calculation

Before

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit3 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, S043

S045 · calculation

Before

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit3 squared base reduction normalized remainder = 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, S044

S046 · implies

Before

Fermat original modular exponent bit3 squared base reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S045

S047 · implies

Before

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[0, 0, 7];low bit=1

After

next[accumulator,base,exponent]=[0, 0, 3];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, S046

S048 · calculation

Before

Fermat original modular exponent bit4 accumulator product := mul(0, 0)

After

Fermat original modular exponent bit4 accumulator product = 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, S047

S049 · calculation

Before

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit4 accumulator reduction exact quotient = 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, S048

S050 · calculation

Before

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(0, 59)

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, S049

S051 · calculation

Before

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

After

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 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, S050

S052 · implies

Before

Fermat original modular exponent bit4 accumulator reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S051

S053 · calculation

Before

Fermat original modular exponent bit4 current base square := mul(0, 0)

After

Fermat original modular exponent bit4 current base square = 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, S052

S054 · calculation

Before

Fermat original modular exponent bit4 squared base reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit4 squared base reduction exact quotient = 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, S053

S055 · calculation

Before

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit4 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, S054

S056 · calculation

Before

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit4 squared base reduction normalized remainder = 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, S055

S057 · implies

Before

Fermat original modular exponent bit4 squared base reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S056

S058 · implies

Before

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[0, 0, 3];low bit=1

After

next[accumulator,base,exponent]=[0, 0, 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, S057

S059 · calculation

Before

Fermat original modular exponent bit5 accumulator product := mul(0, 0)

After

Fermat original modular exponent bit5 accumulator product = 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, S058

S060 · calculation

Before

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit5 accumulator reduction exact quotient = 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, S059

S061 · calculation

Before

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit5 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, S060

S062 · calculation

Before

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 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, S061

S063 · implies

Before

Fermat original modular exponent bit5 accumulator reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S062

S064 · calculation

Before

Fermat original modular exponent bit5 current base square := mul(0, 0)

After

Fermat original modular exponent bit5 current base square = 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, S063

S065 · calculation

Before

Fermat original modular exponent bit5 squared base reduction exact quotient := div(0, 59)

After

Fermat original modular exponent bit5 squared base reduction exact quotient = 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, S064

S066 · calculation

Before

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(0, 59)

After

Fermat original modular exponent bit5 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, S065

S067 · calculation

Before

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(0, 0)

After

Fermat original modular exponent bit5 squared base reduction normalized remainder = 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, S066

S068 · implies

Before

Fermat original modular exponent bit5 squared base reduction:signed integer value=0;positive modulus=59;exact quotient=0;floor q=0

After

value=q*modulus+remainder;normalized remainder=0 in[0,59)

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, S067

S069 · implies

Before

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[0, 0, 1];low bit=1

After

next[accumulator,base,exponent]=[0, 0, 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, S068

S070 · implies

Before

Fermat original modular exponent:remaining exponent0;final normalized accumulator=0

After

EXACT original base^59 modulo59=0

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, S069

S071 · implies

Before

ACTUAL prime p=59;original a=1003 is divisible by p;normalized residue0;positive exponent p

After

a^p residue=0=0=a residue;NO cancellation of a or0 needed

Conditions: none

When p divides a,it also divides the positive p-th power,so both residues are0.This branch does not cancel0 to derive the coprime form.

Step identifiers and dependencies

Rule: flt_all_integer_zero_residue_case_no_invalid_cancellation · Uses: G001, S070

Reverse checks

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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S067 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S066 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S065 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S064 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 current base square with exact rational arithmetic, not rounded decimals.
S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S062 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S061 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 accumulator product with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S056 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 current base square with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S051 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator product with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S045 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 current base square with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S040 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator product with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S034 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 current base square with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S028 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 current base square with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S023 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 accumulator product with exact rational arithmetic, not rounded decimals.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 current base square with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 accumulator product 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

(59, 0, 0, 0, Yes, all_integer_form_including_prime_divides_a)

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: b3d5e6d4d829da47e9ed0db3da6461a1fff7cea708d20a353b4eda6af51b67e3
candidate record SHA-256: 216e0b000c4a588a0d8095eda08730003b517247e584bb60478fe035a5727834
semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
Lean: NOT_RUN · human review: NOT_PERFORMED

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