Who's Who in n^2+1 Primes Research

The n²+1 conjecture asks whether the polynomial n²+1 produces infinitely many prime values: 2, 5, 17, 37, 101, 197, 257, and so on. This is one of Edmund Landau four classical problems, presented at the 1912 International Congress of Mathematicians in Cambridge, alongside Goldbach conjecture, the twin prime conjecture, and Legendre conjecture. The conjecture is still open.

The problem belongs to a broader family: given an irreducible polynomial with integer coefficients and no fixed prime divisor, does it represent infinitely many primes? The general form of this question is Bunyakovsky conjecture (1857), and the simultaneous version for several polynomials is Schinzel Hypothesis H. For n²+1 specifically, Hardy and Littlewood gave in 1923 a precise quantitative prediction for how many primes of this form fall below x, as part of their Conjecture F (a special case of the Bateman-Horn conjecture). The prediction matches computational data closely, yet no proof exists.

The closest results are by Henryk Iwaniec, who proved in 1978 that n²+1 is a product of at most two primes infinitely often (using the half-dimensional sieve), and by John Friedlander and Henryk Iwaniec, who proved in 1997 that the polynomial x²+y⁴ represents infinitely many primes: the first time a sparse two-variable polynomial was shown to be prime-producing. Roger Heath-Brown later extended the method to x³+2y³. Jori Merikoski (2023) proved that n²+1 has a prime factor exceeding n^6/5 infinitely often, a result in the direction of the problem, though not a prime-values result directly.

This site is a reference directory of the researchers most active on the n²+1 conjecture and the surrounding community: polynomial prime values, the Bateman-Horn conjecture, the half-dimensional sieve, and the Friedlander-Iwaniec method. Rankings are built from arXiv preprint output, OpenAlex citation data, and zbMATH MSC classifications (11N32, 11N35, 11N36). Because the problem is specialized, the community is smaller than for Goldbach or twin primes. The corpus reflects that: a focused field.