Formula For Primes - Prime Formulas and Polynomial Functions

Prime Formulas and Polynomial Functions

It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n. The proof is: Suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so . But for any k, also, so (as it is prime and divisible by p), but the only way for all k is if the polynomial function is constant.

The same reasoning shows an even stronger result: no non-constant polynomial function P(n) exists that evaluates to a prime number for almost all integers n.

Euler first noticed (in 1772) that the quadratic polynomial

P(n) = n2 - n + 41

is prime for all positive integers less than 41. The primes for n = 1, 2, 3... are 41, 43, 47, 53, 61, 71... The differences between the terms are 2, 4, 6, 8, 10... For n = 41, it produces a square number, 1681, which is equal to 41×41, the smallest composite number for this formula. If 41 divides n it divides P(n) too. The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number, and there are analogous polynomials for, corresponding to other Heegner numbers.

It is known, based on Dirichlet's theorem on arithmetic progressions, that linear polynomial functions produce infinitely many primes as long as a and b are relatively prime (though no such function will assume prime values for all values of n). Moreover, the Green–Tao theorem says that for any k there exists a pair of a and b with the property that is prime for any n from 0 to k − 1. However, the best known result of such type is for k = 26:

43142746595714191 + 5283234035979900n is prime for all n from 0 to 25 (Andersen 2010).

It is not even known whether there exists a univariate polynomial of degree at least 2 that assumes an infinite number of values that are prime; see Bunyakovsky conjecture.