modulus31

Modulus 31, in number theory, refers to arithmetic performed modulo 31. It uses the residue classes of integers modulo 31, with canonical representatives 0 through 30. Two integers a and b are congruent modulo 31 if their difference a−b is a multiple of 31.

Because 31 is prime, the set of nonzero residues {1, ..., 30} forms a cyclic multiplicative group of order 30 under multiplication modulo 31; each nonzero residue has a multiplicative inverse. The totient φ(31) equals 30, so Fermat's little theorem implies a^30 ≡ 1 (mod 31) for any a not divisible by 31. The group being cyclic means there exist primitive roots modulo 31.

Arithmetic operations are performed with reductions modulo 31. Examples: (9 + 25) ≡ 34 ≡ 3 (mod 31); (9 × 25) ≡ 225 ≡ 8 (mod 31) since 225 − 217 = 8. The multiplicative inverse of 3 modulo 31 is 21, because 3 × 21 ≡ 1 (mod 31). Inverse elements exist for all nonzero residues due to the prime modulus.

Applications and context: modulus 31 arithmetic is foundational in teaching modular arithmetic and is a component in various computational contexts, including cryptographic protocols that rely on properties of larger prime moduli, hash functions, and error-detection schemes. The primality of 31 affords convenient inverses and predictable exponent behavior via Fermat’s theorem. Notably, 31 is also a Mersenne prime, equal to 2^5 − 1, highlighting its historical relevance in prime number theory.

In summary, modulus 31 describes the arithmetic of residues modulo 31, with a rich algebraic structure and broad educational and computational relevance.