# Question 19899: Use the Prime-Coprime Fermat Form Only under Both Hypotheses [Return to catalogue page 100](https://ocme.evemisslab.com/en/questions/page/100/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations/#q-20261014-advance-20261106-r05-fermat-little-theorem-0024) 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. R0038-19899 · q-20261014-advance-20261106-r05-fermat-little-theorem-0024 ## Original question and givens Use integer base a=-25 and modulus=89, 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_minus_1, congruence_certificate, boundary). - G001 — Use integer base a=-25 and modulus=89, 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_minus_1, congruence_certificate, 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](https://ocme.evemisslab.com/en/mko/mko-fermat-little-theorem/) ## Step-by-step derivation S001 · impliesBefore source operation=flt-prime-coprime;original snapshot and mathematics retained After explicit ordered source output fields=[p, normalized_base, power_p_minus_1, congruence_certificate, 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ACTUAL integer modulus=89;complete trial range2..floor_sqrt=9 After checked rows[divisor,quotient,remainder]=[[2, 44, 1], [3, 29, 2], [4, 22, 1], [5, 17, 4], [6, 14, 5], [7, 12, 5], [8, 11, 1], [9, 9, 8]];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 dependenciesRule: flt_primality_complete_trial_or_actual_factor_not_Fermat_test · Uses: G001, S001 S003 · impliesBefore original signed base=-25;positive modulus=89;use nonnegative gcd inputs=[25, 89] After ALL Euclidean rows[left,right,q,r]=[[25, 89, 0, 25], [89, 25, 3, 14], [25, 14, 1, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=Yes Conditions: none Euclidean replacements preserve common divisors and give the terminal gcd,including gcd(0,m)=m.Primality of the modulus and invertibility of the base remain separate flags. Step identifiers and dependenciesRule: flt_actual_gcd_coprime_separate_from_primality · Uses: G001, S002 S004 · calculationBefore Fermat original signed base exact quotient := div(-25, 89) After Fermat original signed base exact quotient = -25/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Fermat original signed base floor quotient multiple := mul(-1, 89) After Fermat original signed base floor quotient multiple = -89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Fermat original signed base normalized remainder := sub(-25, -89) After Fermat original signed base normalized remainder = 64 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore Fermat original signed base:signed integer value=-25;positive modulus=89;exact quotient=-25/89;floor q=-1 After value=q*modulus+remainder;normalized remainder=64 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S006 S008 · impliesBefore Fermat original modular exponent:original normalized base=64;original exponent=88;modulus=89 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 dependenciesRule: flt_modular_power_initial_invariant · Uses: G001, S007 S009 · calculationBefore Fermat original modular exponent bit0 current base square := mul(64, 64) After Fermat original modular exponent bit0 current base square = 4096 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Fermat original modular exponent bit0 squared base reduction exact quotient := div(4096, 89) After Fermat original modular exponent bit0 squared base reduction exact quotient = 4096/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(46, 89) After Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 4094 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(4096, 4094) After Fermat original modular exponent bit0 squared base reduction normalized remainder = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore Fermat original modular exponent bit0 squared base reduction:signed integer value=4096;positive modulus=89;exact quotient=4096/89;floor q=46 After value=q*modulus+remainder;normalized remainder=2 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S012 S014 · impliesBefore Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 64, 88];low bit=0 After next[accumulator,base,exponent]=[1, 2, 44];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 dependenciesRule: flt_each_binary_power_step_exact_invariant · Uses: G001, S013 S015 · calculationBefore Fermat original modular exponent bit1 current base square := mul(2, 2) After Fermat original modular exponent bit1 current base square = 4 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Fermat original modular exponent bit1 squared base reduction exact quotient := div(4, 89) After Fermat original modular exponent bit1 squared base reduction exact quotient = 4/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 89) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(4, 0) After Fermat original modular exponent bit1 squared base reduction normalized remainder = 4 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore Fermat original modular exponent bit1 squared base reduction:signed integer value=4;positive modulus=89;exact quotient=4/89;floor q=0 After value=q*modulus+remainder;normalized remainder=4 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S018 S020 · impliesBefore Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 2, 44];low bit=0 After next[accumulator,base,exponent]=[1, 4, 22];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 dependenciesRule: flt_each_binary_power_step_exact_invariant · Uses: G001, S019 S021 · calculationBefore Fermat original modular exponent bit2 current base square := mul(4, 4) After Fermat original modular exponent bit2 current base square = 16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Fermat original modular exponent bit2 squared base reduction exact quotient := div(16, 89) After Fermat original modular exponent bit2 squared base reduction exact quotient = 16/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(0, 89) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(16, 0) After Fermat original modular exponent bit2 squared base reduction normalized remainder = 16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore Fermat original modular exponent bit2 squared base reduction:signed integer value=16;positive modulus=89;exact quotient=16/89;floor q=0 After value=q*modulus+remainder;normalized remainder=16 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S024 S026 · impliesBefore Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 4, 22];low bit=0 After next[accumulator,base,exponent]=[1, 16, 11];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 dependenciesRule: flt_each_binary_power_step_exact_invariant · Uses: G001, S025 S027 · calculationBefore Fermat original modular exponent bit3 accumulator product := mul(1, 16) After Fermat original modular exponent bit3 accumulator product = 16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Fermat original modular exponent bit3 accumulator reduction exact quotient := div(16, 89) After Fermat original modular exponent bit3 accumulator reduction exact quotient = 16/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(0, 89) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(16, 0) After Fermat original modular exponent bit3 accumulator reduction normalized remainder = 16 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore Fermat original modular exponent bit3 accumulator reduction:signed integer value=16;positive modulus=89;exact quotient=16/89;floor q=0 After value=q*modulus+remainder;normalized remainder=16 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S030 S032 · calculationBefore Fermat original modular exponent bit3 current base square := mul(16, 16) After Fermat original modular exponent bit3 current base square = 256 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Fermat original modular exponent bit3 squared base reduction exact quotient := div(256, 89) After Fermat original modular exponent bit3 squared base reduction exact quotient = 256/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(2, 89) After Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 178 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(256, 178) After Fermat original modular exponent bit3 squared base reduction normalized remainder = 78 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · impliesBefore Fermat original modular exponent bit3 squared base reduction:signed integer value=256;positive modulus=89;exact quotient=256/89;floor q=2 After value=q*modulus+remainder;normalized remainder=78 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S035 S037 · impliesBefore Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 16, 11];low bit=1 After next[accumulator,base,exponent]=[16, 78, 5];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 dependenciesRule: flt_each_binary_power_step_exact_invariant · Uses: G001, S036 S038 · calculationBefore Fermat original modular exponent bit4 accumulator product := mul(16, 78) After Fermat original modular exponent bit4 accumulator product = 1248 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Fermat original modular exponent bit4 accumulator reduction exact quotient := div(1248, 89) After Fermat original modular exponent bit4 accumulator reduction exact quotient = 1248/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(14, 89) After Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 1246 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(1248, 1246) After Fermat original modular exponent bit4 accumulator reduction normalized remainder = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore Fermat original modular exponent bit4 accumulator reduction:signed integer value=1248;positive modulus=89;exact quotient=1248/89;floor q=14 After value=q*modulus+remainder;normalized remainder=2 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S041 S043 · calculationBefore Fermat original modular exponent bit4 current base square := mul(78, 78) After Fermat original modular exponent bit4 current base square = 6084 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Fermat original modular exponent bit4 squared base reduction exact quotient := div(6084, 89) After Fermat original modular exponent bit4 squared base reduction exact quotient = 6084/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(68, 89) After Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 6052 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(6084, 6052) After Fermat original modular exponent bit4 squared base reduction normalized remainder = 32 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore Fermat original modular exponent bit4 squared base reduction:signed integer value=6084;positive modulus=89;exact quotient=6084/89;floor q=68 After value=q*modulus+remainder;normalized remainder=32 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S046 S048 · impliesBefore Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[16, 78, 5];low bit=1 After next[accumulator,base,exponent]=[2, 32, 2];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 dependenciesRule: flt_each_binary_power_step_exact_invariant · Uses: G001, S047 S049 · calculationBefore Fermat original modular exponent bit5 current base square := mul(32, 32) After Fermat original modular exponent bit5 current base square = 1024 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Fermat original modular exponent bit5 squared base reduction exact quotient := div(1024, 89) After Fermat original modular exponent bit5 squared base reduction exact quotient = 1024/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(11, 89) After Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 979 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(1024, 979) After Fermat original modular exponent bit5 squared base reduction normalized remainder = 45 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · impliesBefore Fermat original modular exponent bit5 squared base reduction:signed integer value=1024;positive modulus=89;exact quotient=1024/89;floor q=11 After value=q*modulus+remainder;normalized remainder=45 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S052 S054 · impliesBefore Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[2, 32, 2];low bit=0 After next[accumulator,base,exponent]=[2, 45, 1];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 dependenciesRule: flt_each_binary_power_step_exact_invariant · Uses: G001, S053 S055 · calculationBefore Fermat original modular exponent bit6 accumulator product := mul(2, 45) After Fermat original modular exponent bit6 accumulator product = 90 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Fermat original modular exponent bit6 accumulator reduction exact quotient := div(90, 89) After Fermat original modular exponent bit6 accumulator reduction exact quotient = 90/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Fermat original modular exponent bit6 accumulator reduction floor quotient multiple := mul(1, 89) After Fermat original modular exponent bit6 accumulator reduction floor quotient multiple = 89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Fermat original modular exponent bit6 accumulator reduction normalized remainder := sub(90, 89) After Fermat original modular exponent bit6 accumulator reduction normalized remainder = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · impliesBefore Fermat original modular exponent bit6 accumulator reduction:signed integer value=90;positive modulus=89;exact quotient=90/89;floor q=1 After value=q*modulus+remainder;normalized remainder=1 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S058 S060 · calculationBefore Fermat original modular exponent bit6 current base square := mul(45, 45) After Fermat original modular exponent bit6 current base square = 2025 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore Fermat original modular exponent bit6 squared base reduction exact quotient := div(2025, 89) After Fermat original modular exponent bit6 squared base reduction exact quotient = 2025/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(22, 89) After Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 1958 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(2025, 1958) After Fermat original modular exponent bit6 squared base reduction normalized remainder = 67 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · impliesBefore Fermat original modular exponent bit6 squared base reduction:signed integer value=2025;positive modulus=89;exact quotient=2025/89;floor q=22 After value=q*modulus+remainder;normalized remainder=67 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S063 S065 · impliesBefore Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[2, 45, 1];low bit=1 After next[accumulator,base,exponent]=[1, 67, 0];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 dependenciesRule: flt_each_binary_power_step_exact_invariant · Uses: G001, S064 S066 · impliesBefore Fermat original modular exponent:remaining exponent0;final normalized accumulator=1 After EXACT original base^88 modulo89=1 Conditions: none At remaining exponent0 the invariant leaves exactly the accumulator,giving the original power's residue.The arithmetic result alone does not certify a prime modulus. Step identifiers and dependenciesRule: flt_modular_power_terminal_not_primality_converse · Uses: G001, S065 S067 · calculationBefore Fermat actual unit inverse product := mul(64, 32) After Fermat actual unit inverse product = 2048 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore Fermat actual inverse identity exact quotient := div(2048, 89) After Fermat actual inverse identity exact quotient = 2048/89 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore Fermat actual inverse identity floor quotient multiple := mul(23, 89) After Fermat actual inverse identity floor quotient multiple = 2047 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore Fermat actual inverse identity normalized remainder := sub(2048, 2047) After Fermat actual inverse identity normalized remainder = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · impliesBefore Fermat actual inverse identity:signed integer value=2048;positive modulus=89;exact quotient=2048/89;floor q=23 After value=q*modulus+remainder;normalized remainder=1 in[0,89) 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 dependenciesRule: flt_exact_signed_modulo_not_integer_equality · Uses: G001, S070 S072 · impliesBefore VERIFIED prime p=89;nonzero original base residue=64;ACTUAL inverse witness=32 After multiplication r->a*r mod p permutes ALL nonzero residues1..p-1:inverse gives injectivity and finite self-injection gives surjectivity Conditions: none The actual inverse permits cancellation,so multiplication by a is a bijection of the full nonzero residue set.This is an argument for arbitrary residues,not a million-point enumeration or sampled permutation test. Step identifiers and dependenciesRule: flt_prime_unit_permutation_all_residues_not_samples · Uses: G001, S071 S073 · impliesBefore same verified prime p=89;ALL nonzero residues permuted by multiplication by a=64 After product over r=1..p-1 gives a^(p-1)*(p-1)! congruent to (p-1)!;EVERY factor is a prime-field unit,so factorial is a UNIT;cancel ONLY this nonzero unit to obtain a^(p-1) congruent1 Conditions: none For a prime modulus every factor1 through p−1 is coprime to p and has an inverse;their finite product is invertible as well.Cancel that factorial unit to obtain Fermat's coprime form.Never cancel0 or apply this argument to nonunits under a composite modulus. Step identifiers and dependenciesRule: flt_nonzero_factorial_unit_cancellation_not_zero_or_composite · Uses: G001, S072 ## Reverse checks - S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S070 · premise_and_result_audit — Recompute Fermat actual inverse identity normalized remainder with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Fermat actual inverse identity floor quotient multiple with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute Fermat actual inverse identity exact quotient with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute Fermat actual unit inverse product with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 current base square with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Fermat original modular exponent bit6 accumulator product with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Fermat original modular exponent bit5 current base square with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 current base square with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Fermat original modular exponent bit4 accumulator product with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 current base square with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Fermat original modular exponent bit3 accumulator product with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Fermat original modular exponent bit2 current base square with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Fermat original modular exponent bit1 current base square with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction normalized remainder with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 squared base reduction exact quotient with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Fermat original modular exponent bit0 current base square with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute Fermat original signed base normalized remainder with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Fermat original signed base floor quotient multiple with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Fermat original signed base exact quotient with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (89, 64, 1, Yes, prime_coprime_form) 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: 347ce627d33090206064d66f3351fec33a02c0f4698957824fbaf832f136dd12 - candidate record SHA-256: 7987e07bf12d69b0d23a555b3f8aebeb85fbbdbe8a5f43fb29759882ee826475 - semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612 - question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r05-fermat-little-theorem-0024.json)