Landau 4: primes of the form n^2+1

• Slug: landau-n2-plus-1
• List: Landau 4 (1912); also Hilbert 8
• Solver: Maths + SuperGrok CLI grok-4.6 xhigh
• Status: open; dent: certified $n\le 10^8$ matching Wolf $\pi_q(10^{16})$
• Area: Analytic / computational number theory
• Sources: Landau ICM 1912; Iwaniec 1978; OEIS A002496; Wolf arXiv:0803.1456
• Started: 2026-08-17

Statement

Are there infinitely many primes of the form n^2+1?

Related but different: Landau 3, primes between consecutive squares.

Tonight

Do not claim infinitude. A new bound is a replayable finite record:

• an independently certified list of n with n^2+1 prime, up to a stated N
• the Iwaniec P2s in the same range (at most two prime factors, multiplicity)
• or a prime n^2+1 that is not on the published list we cite

Incomplete search is not a lower bound on the number of such primes.

Done for N=10^8: 3954181 primes (matches Wolf π_q(10^16)), 12172983 Ω=2 composites, Bateman–Horn ratio 0.99972. Extends the N=10^7 prefix (456362 primes, Wolf π_q(10^14)). No new prime off Wolf/Grantham.