# Question 20058: A Nonunit Fails Euler Applicability, Not a Primality Test

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

R0039-20058 · q-20261014-advance-20261107-r06-euler-theorem-number-theory-0083



## Original question and givens

Use integer modulus n=13≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).


- G001 — Use integer modulus n=13≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, 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-euler-theorem-number-theory](https://ocme.evemisslab.com/en/mko/mko-euler-theorem-number-theory/)





## Step-by-step derivation

S001 · impliesBefore

source operation=eulerthm-nonunit-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, original_base, gcd, theorem_applicable, residue, boundary]

Conditions: none

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

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · calculationBefore

Euler ORIGINAL signed base normalization exact quotient := div(13, 13)

After

Euler ORIGINAL signed base normalization exact quotient = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S003 · calculationBefore

Euler ORIGINAL signed base normalization floor quotient multiple := mul(1, 13)

After

Euler ORIGINAL signed base normalization floor quotient multiple = 13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S004 · calculationBefore

Euler ORIGINAL signed base normalization normalized remainder := sub(13, 13)

After

Euler ORIGINAL signed base normalization normalized remainder = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S005 · impliesBefore

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=13;modulus=13;exact ratio=1;floor quotient=1

After

value=q*n+remainder;normalized residue=0 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

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





S006 · impliesBefore

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[13, 13];nonnegative initial pair=[13, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[13, 13, 1, 0]];terminal gcd=13

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S007 · impliesBefore

Euler residue representative0:ORIGINAL gcd inputs=[0, 13];nonnegative initial pair=[0, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[0, 13, 0, 0]];terminal gcd=13

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S008 · impliesBefore

Euler residue representative1:ORIGINAL gcd inputs=[1, 13];nonnegative initial pair=[1, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[1, 13, 0, 1], [13, 1, 13, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S009 · impliesBefore

Euler residue representative2:ORIGINAL gcd inputs=[2, 13];nonnegative initial pair=[2, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[2, 13, 0, 2], [13, 2, 6, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S010 · impliesBefore

Euler residue representative3:ORIGINAL gcd inputs=[3, 13];nonnegative initial pair=[3, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[3, 13, 0, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S011 · impliesBefore

Euler residue representative4:ORIGINAL gcd inputs=[4, 13];nonnegative initial pair=[4, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[4, 13, 0, 4], [13, 4, 3, 1], [4, 1, 4, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S012 · impliesBefore

Euler residue representative5:ORIGINAL gcd inputs=[5, 13];nonnegative initial pair=[5, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[5, 13, 0, 5], [13, 5, 2, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S013 · impliesBefore

Euler residue representative6:ORIGINAL gcd inputs=[6, 13];nonnegative initial pair=[6, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[6, 13, 0, 6], [13, 6, 2, 1], [6, 1, 6, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S014 · impliesBefore

Euler residue representative7:ORIGINAL gcd inputs=[7, 13];nonnegative initial pair=[7, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[7, 13, 0, 7], [13, 7, 1, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S015 · impliesBefore

Euler residue representative8:ORIGINAL gcd inputs=[8, 13];nonnegative initial pair=[8, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[8, 13, 0, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S016 · impliesBefore

Euler residue representative9:ORIGINAL gcd inputs=[9, 13];nonnegative initial pair=[9, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[9, 13, 0, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S017 · impliesBefore

Euler residue representative10:ORIGINAL gcd inputs=[10, 13];nonnegative initial pair=[10, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[10, 13, 0, 10], [13, 10, 1, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S018 · impliesBefore

Euler residue representative11:ORIGINAL gcd inputs=[11, 13];nonnegative initial pair=[11, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[11, 13, 0, 11], [13, 11, 1, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S019 · impliesBefore

Euler residue representative12:ORIGINAL gcd inputs=[12, 13];nonnegative initial pair=[12, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[12, 13, 0, 12], [13, 12, 1, 1], [12, 1, 12, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

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





S020 · calculationBefore

Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

After

Euler complete reduced residue count phi = 12

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S021 · impliesBefore

COMPLETE original residue carrier0..12;ALL [representative,gcd,unit] rows=[[0, 13, No], [1, 1, Yes], [2, 1, Yes], [3, 1, Yes], [4, 1, Yes], [5, 1, Yes], [6, 1, Yes], [7, 1, Yes], [8, 1, Yes], [9, 1, Yes], [10, 1, Yes], [11, 1, Yes], [12, 1, Yes]]

After

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];phi(n)=12;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Conditions: none

Test gcd1 for every representative to obtain the complete unit set,which contains1.The source totient count uses1..n while the unit list uses0..n−1.For n≥2 both swapped endpoints0 and n are nonunits,so the counts agree without assuming a prime modulus.

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





S022 · impliesBefore

ORIGINAL normalized base=0;COMPLETE totient exponent=12;modulus=13

After

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

Conditions: none

Use finite square-and-multiply steps preserving the original totient-power congruence,without constructing huge ordinary powers.Phi is the complete unit count,not n−1 unless primality is separately established.

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





S023 · calculationBefore

Euler power bit0 base square := mul(0, 0)

After

Euler power bit0 base square = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S024 · calculationBefore

Euler power bit0 squared-base reduction exact quotient := div(0, 13)

After

Euler power bit0 squared-base reduction exact quotient = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S025 · calculationBefore

Euler power bit0 squared-base reduction floor quotient multiple := mul(0, 13)

After

Euler power bit0 squared-base reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S026 · calculationBefore

Euler power bit0 squared-base reduction normalized remainder := sub(0, 0)

After

Euler power bit0 squared-base reduction normalized remainder = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S027 · impliesBefore

Euler power bit0 squared-base reduction:ORIGINAL signed integer=0;modulus=13;exact ratio=0;floor quotient=0

After

value=q*n+remainder;normalized residue=0 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

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





S028 · impliesBefore

step=0;previous[accumulator,base,exponent]=[1, 0, 12];low bit=0

After

next[accumulator,base,exponent]=[1, 0, 6];split exponent=2*q+bit preserves original-power invariant

Conditions: none

For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability.

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





S029 · calculationBefore

Euler power bit1 base square := mul(0, 0)

After

Euler power bit1 base square = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S030 · calculationBefore

Euler power bit1 squared-base reduction exact quotient := div(0, 13)

After

Euler power bit1 squared-base reduction exact quotient = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S031 · calculationBefore

Euler power bit1 squared-base reduction floor quotient multiple := mul(0, 13)

After

Euler power bit1 squared-base reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S032 · calculationBefore

Euler power bit1 squared-base reduction normalized remainder := sub(0, 0)

After

Euler power bit1 squared-base reduction normalized remainder = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S033 · impliesBefore

Euler power bit1 squared-base reduction:ORIGINAL signed integer=0;modulus=13;exact ratio=0;floor quotient=0

After

value=q*n+remainder;normalized residue=0 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

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





S034 · impliesBefore

step=1;previous[accumulator,base,exponent]=[1, 0, 6];low bit=0

After

next[accumulator,base,exponent]=[1, 0, 3];split exponent=2*q+bit preserves original-power invariant

Conditions: none

For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability.

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





S035 · calculationBefore

Euler power bit2 accumulator product := mul(1, 0)

After

Euler power bit2 accumulator product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S036 · calculationBefore

Euler power bit2 accumulator reduction exact quotient := div(0, 13)

After

Euler power bit2 accumulator reduction exact quotient = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S037 · calculationBefore

Euler power bit2 accumulator reduction floor quotient multiple := mul(0, 13)

After

Euler power bit2 accumulator reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S038 · calculationBefore

Euler power bit2 accumulator reduction normalized remainder := sub(0, 0)

After

Euler power bit2 accumulator reduction normalized remainder = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S039 · impliesBefore

Euler power bit2 accumulator reduction:ORIGINAL signed integer=0;modulus=13;exact ratio=0;floor quotient=0

After

value=q*n+remainder;normalized residue=0 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

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





S040 · calculationBefore

Euler power bit2 base square := mul(0, 0)

After

Euler power bit2 base square = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S041 · calculationBefore

Euler power bit2 squared-base reduction exact quotient := div(0, 13)

After

Euler power bit2 squared-base reduction exact quotient = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S042 · calculationBefore

Euler power bit2 squared-base reduction floor quotient multiple := mul(0, 13)

After

Euler power bit2 squared-base reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S043 · calculationBefore

Euler power bit2 squared-base reduction normalized remainder := sub(0, 0)

After

Euler power bit2 squared-base reduction normalized remainder = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S044 · impliesBefore

Euler power bit2 squared-base reduction:ORIGINAL signed integer=0;modulus=13;exact ratio=0;floor quotient=0

After

value=q*n+remainder;normalized residue=0 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

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





S045 · impliesBefore

step=2;previous[accumulator,base,exponent]=[1, 0, 3];low bit=1

After

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

Conditions: none

For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability.

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





S046 · calculationBefore

Euler power bit3 accumulator product := mul(0, 0)

After

Euler power bit3 accumulator product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S047 · calculationBefore

Euler power bit3 accumulator reduction exact quotient := div(0, 13)

After

Euler power bit3 accumulator reduction exact quotient = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S048 · calculationBefore

Euler power bit3 accumulator reduction floor quotient multiple := mul(0, 13)

After

Euler power bit3 accumulator reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S049 · calculationBefore

Euler power bit3 accumulator reduction normalized remainder := sub(0, 0)

After

Euler power bit3 accumulator reduction normalized remainder = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S050 · impliesBefore

Euler power bit3 accumulator reduction:ORIGINAL signed integer=0;modulus=13;exact ratio=0;floor quotient=0

After

value=q*n+remainder;normalized residue=0 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

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





S051 · calculationBefore

Euler power bit3 base square := mul(0, 0)

After

Euler power bit3 base square = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S052 · calculationBefore

Euler power bit3 squared-base reduction exact quotient := div(0, 13)

After

Euler power bit3 squared-base reduction exact quotient = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S053 · calculationBefore

Euler power bit3 squared-base reduction floor quotient multiple := mul(0, 13)

After

Euler power bit3 squared-base reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S054 · calculationBefore

Euler power bit3 squared-base reduction normalized remainder := sub(0, 0)

After

Euler power bit3 squared-base reduction normalized remainder = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

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





S055 · impliesBefore

Euler power bit3 squared-base reduction:ORIGINAL signed integer=0;modulus=13;exact ratio=0;floor quotient=0

After

value=q*n+remainder;normalized residue=0 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

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





S056 · impliesBefore

step=3;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

For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability.

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





S057 · impliesBefore

remaining exponent0;final normalized accumulator=0

After

EXACT original normalized base^12 modulo13=0

Conditions: none

The terminal invariant gives the original totient-power residue.This is arithmetic modulo the stated modulus,not a primality certificate.

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





S058 · impliesBefore

ACTUAL original n=13,a=13;gcd=13>1;phi(n)=12;computed totient-power residue=0

After

Euler theorem applicability=No because gcd1 premise fails;retain ACTUAL residue;do NOT cancel a nonunit or infer prime/composite status from this marker

Conditions: none

The No rejects the gcd1 premise without forcing the actual residue to1 or classifying primality.The modulus itself may be prime or composite.Fermat's all-integer form is different from the coprimality-dependent claim a^phi≡1.

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









## Reverse checks


- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S054 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S053 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Recompute Euler power bit3 base square with exact rational arithmetic, not rounded decimals.
- S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S049 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Recompute Euler power bit3 accumulator product with exact rational arithmetic, not rounded decimals.
- S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S041 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Recompute Euler power bit2 base square with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S038 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S037 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Recompute Euler power bit2 accumulator product with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S032 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S029 · premise_and_result_audit — Recompute Euler power bit1 base square with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S025 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S023 · premise_and_result_audit — Recompute Euler power bit0 base square with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S020 · premise_and_result_audit — Recompute Euler complete reduced residue count phi 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization normalized remainder with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization exact quotient with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(13, 13, 13, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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: 5bf8877a2a290f7875a714e07dd25309273f06adeb0e2117ee55186398919fa3
- candidate record SHA-256: 101dee239308e3470e73395cf399ed729a1beadb9b4d4e98b6dd03302485aabd
- semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034
- question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0083.json)