Fermat's little theorem checker

This Fermat's little theorem checker takes integers a and n, refuses the pair when gcd(a, n) is not one, and otherwise returns a^(n−1) mod n with a boolean flag telling you whether the remainder is one. The same pure BigInt logic powers the free browser tool and the paid API path at $0.002 per successful request, so classroom demos and production scripts never disagree on a residue.

What Fermat's little theorem claims and what this tool computes

The proof is not the product of this endpoint; the product is the exact remainder of that power. The response echoes a and n as decimal strings, reports the exponent n−1, the non-negative remainder result, gcd as the string "1", a boolean congruent_to_one that is true exactly when result equals 1, and a short expression string you can print next to the answer. Classic prime check: a equals 2 and n equals 7 yields 2^6 mod 7 equals 1, so congruent_to_one is true. Classic composite probe: a equals 2 and n equals 15 yields 2^14 mod 15 equals 4, so congruent_to_one is false—exactly the kind of counter-example that shows a^(n−1) ≡ 1 is not a primality certificate by itself.

Input contract, errors, and how to read the response

If n is less than two the task fails with a message that the modulus must be at least two. If gcd(a, n) is not one the task fails with a message that names the gcd—for example gcd(2, 4) equals 2—so callers know the pair is outside the coprime hypothesis rather than guessing why the power was not computed. Negative bases are accepted and reduced into the non-negative residue system modulo n before powering, which matches standard modular arithmetic. POST a JSON body with two required integer fields: a (aliases base) and n (aliases modulus, mod). Each field may be a JSON number or a decimal integer string; string form is preferred when values leave JavaScript's safe integer range. Fractions, empty strings, non-numeric tokens, and values longer than sixty-four decimal digits fail with invalid_input and are never billed. On success every numeric field in the body is a decimal string except congruent_to_one, which is a boolean, so large integers round-trip through JSON without precision loss. Pair this checker with a separate factoring or primality tool when your workflow needs more than the power residue; this endpoint answers only a^(n−1) mod n under the coprime gate so the SEO intent stays focused and the response stays minimal. Pricing is a flat $0.002 per successful request on the item unit; failed validation does not consume prepaid balance.

Classroom demos, composite probes, and API automation

Teachers use a Fermat little theorem checker so students can verify hand calculations of a^(p−1) mod p for small primes before they write a proof, and so they can deliberately try composite n and see that the remainder is not always one. Exam generators and LMS plugins need a deterministic oracle that rejects non-coprime input instead of silently returning a remainder that would confuse graders. Crypto-course notebooks often need a single exact modular power step when introducing RSA toy keys or illustrating why the exponent n−1 appears in elementary number theory before Euler's theorem generalizes the story with φ(n). Developers wire the API into property tests that assert a^(n−1) ≡ 1 for random bases coprime to a known prime modulus, or into regression suites that pin known composite counter-examples such as 2^14 mod 15 equals 4. Keep a and n within the published digit limit, pin the string-form result and congruent_to_one flag in your tests, and you get reproducible residues for homework, tutoring platforms, and edge workers without standing up a full computer-algebra system. When you later need Euler's theorem with φ(n), modular inverses, or Chinese remainder decompositions, call those separate capabilities; this one stays the dedicated Fermat a^(n−1) mod n landing so search intent and response shape remain clear.