Question 19801: The All-Integer Prime Form Includes Bases Divisible by p

Original question and givens

Use integer base a=2 and modulus=2, 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=2 and modulus=2, 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

source operation=flt-all-integer;original snapshot and mathematics retained

explicit ordered source output fields=[p, normalized_base, power_p, comparison_residue, congruent, boundary]

ACTUAL integer modulus=2;complete trial range2..floor_sqrt=1

checked rows[divisor,quotient,remainder]=[];first exact factor=null;prime=Yes

original signed base=2;positive modulus=2;use nonnegative gcd inputs=[2, 2]

ALL Euclidean rows[left,right,q,r]=[[2, 2, 1, 0]];terminal nonnegative gcd=2;coprime=No

Fermat original signed base exact quotient := div(2, 2)

Fermat original signed base exact quotient = 1

Fermat original signed base floor quotient multiple := mul(1, 2)

Fermat original signed base floor quotient multiple = 2

Fermat original signed base normalized remainder := sub(2, 2)

Fermat original signed base normalized remainder = 0

Fermat original signed base:signed integer value=2;positive modulus=2;exact quotient=1;floor q=1

value=q*modulus+remainder;normalized remainder=0 in[0,2)

Fermat original modular exponent:original normalized base=0;original exponent=2;modulus=2

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

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

Fermat original modular exponent bit0 current base square = 0

Fermat original modular exponent bit0 squared base reduction exact quotient := div(0, 2)

Fermat original modular exponent bit0 squared base reduction exact quotient = 0

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(0, 2)

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

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

Fermat original modular exponent bit0 squared base reduction normalized remainder = 0

Fermat original modular exponent bit0 squared base reduction:signed integer value=0;positive modulus=2;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,2)

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

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

Fermat original modular exponent bit1 accumulator product := mul(1, 0)

Fermat original modular exponent bit1 accumulator product = 0

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(0, 2)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 0

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 2)

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple = 0

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 0

Fermat original modular exponent bit1 accumulator reduction:signed integer value=0;positive modulus=2;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,2)

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

Fermat original modular exponent bit1 current base square = 0

Fermat original modular exponent bit1 squared base reduction exact quotient := div(0, 2)

Fermat original modular exponent bit1 squared base reduction exact quotient = 0

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 2)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 0

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

Fermat original modular exponent bit1 squared base reduction normalized remainder = 0

Fermat original modular exponent bit1 squared base reduction:signed integer value=0;positive modulus=2;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,2)

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

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

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

EXACT original base^2 modulo2=0

ACTUAL prime p=2;original a=2 is divisible by p;normalized residue0;positive exponent p

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

Reverse checks

• 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 — 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 — 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 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 current base square 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 — Recompute Fermat original modular exponent bit1 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 accumulator product 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

(2, 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.