Primegenerating

Primegenerating is a term used in number theory to describe methods, polynomials, or sequences that produce prime numbers. The study focuses on how to construct or identify outputs that are prime, and on the limits of such constructions. It often compares explicit formulas, algorithmic procedures, and probabilistic techniques that yield primes or verify primality.

Among polynomials with integer coefficients, certain low-degree examples generate primes for many consecutive inputs. The best known is Euler's polynomial f(n) = n^2 - n + 41, which yields a prime for every integer n with 0 ≤ n ≤ 39, and for some nearby values. Another example is f(n) = n^2 + n + 41, which also yields many primes in an initial block. In general, no non-constant integer polynomial can yield primes for every integer input; after a point, many values become composite. The question of whether a polynomial can produce infinitely many primes is open in general; Bunyakovsky's conjecture posits that irreducible polynomials with positive leading coefficient produce infinitely many primes, but this remains unproven.

Sieve-based methods form a central class of prime-generating procedures. The Sieve of Eratosthenes generates all primes up to a bound by successively marking multiples of discovered primes. Modern sieves and probabilistic primality tests enable rapid generation or verification of large primes, which is essential for cryptographic applications. Pseudorandom prime generators use probabilistic tests, such as Miller–Rabin, to produce candidate primes until a true prime is found.

In practice, prime-generating capabilities span theoretical constructions, algorithmic sieves, and random-prime generation, reflecting both the patterns primes exhibit and the computational tools used to find them.