Fermat's Little Theorem Calculator
Fermat's little theorem calculator is here to help you understand what this famous theorem says and how to use it cor...
What Fermat's Little Theorem Calculator does
Fermat's Little Theorem Calculator on Omni Calculator helps users verify whether a given statement conforms to the theorem or explore its implications for specific numbers. The tool explains that if p is a prime number and a is an integer, then a^p – a is divisible by p. If a is not divisible by p, then a^(p-1) ≡ 1 (mod p). The calculator provides a practical interface for testing primality and finding multiplicative inverses modulo p. It also links to related tools like the modulo calculator and relatively prime calculator for users needing to refresh foundational concepts or verify the coprime condition. The site presents the theorem's history and applications, making it accessible for both learning and practical verification.
How to use the Omni Calculator Fermat's Little Theorem Calculator
- 1
Input the value of 'a' (the integer) and 'p' (the prime number) into the designated fields
- 2
The calculator checks if a and p are coprime by computing their greatest common factor
- 3
Review the result showing whether a^(p-1) ≡ 1 (mod p) holds true
- 4
Use the output to verify primality or find the multiplicative inverse modulo p
- 5
Consult the linked modulo or relatively prime calculators for additional support with modular arithmetic concepts
Best for
Students and enthusiasts of number theory who want to verify Fermat's Little Theorem for specific numbers or learn how to perform primality tests and find multiplicative inverses modulo a prime.
Limitations
- Results depend on the user correctly identifying 'p' as a prime number
- The tool provides estimates based on modular arithmetic and does not prove primality definitively
- No unit switching or conversion between different modular systems is available
Fermat's Little Theorem Calculator FAQ
- What does Fermat's Little Theorem actually state?
- Fermat's Little Theorem states that if p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p). In simpler terms, raising any integer to the power of one less than a prime, then dividing by that prime, always leaves a remainder of 1.
- How can I use this calculator to test if a number is prime?
- To test primality, choose a random integer 'a' that is not divisible by 'p'. If the calculator shows a^(p-1) ≡ 1 (mod p) holds true, 'p' is likely prime. However, this test can produce false positives for certain composite numbers known as Carmichael numbers, so it is not definitive proof of primality.
- What does it mean when the calculator says 'a and p are coprime'?
- Two numbers are coprime if their greatest common factor is 1. The calculator helps verify this condition; if a and p share any common factor other than 1, the theorem's specific form (a^(p-1) ≡ 1 mod p) does not apply, and you should use the general form a^p ≡ a (mod p) instead.
- Can I use Fermat's Little Theorem to find the multiplicative inverse modulo p?
- Yes. If p is prime and a is not divisible by p, then the multiplicative inverse of a modulo p is a^(p-2) mod p. The calculator demonstrates this application, showing how raising a to the power of (p-2) and reducing modulo p yields the value that, when multiplied by a, gives a result of 1 modulo p.
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