What does this research mean for the field?

Primes of the form n^2+1 distribute evenly across the 6N±1 channels without Chebyshev-type bias, and their empirical count up to n=3×10^7 matches the Bateman-Horn heuristic and Landau-Shanks constant to within 0.02%. Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

What question did this study set out to answer?

This study investigates the distribution of primes of the form n^2+1 and validates the Landau–Shanks constant as a measure of their density.

June 10, 2026Open Access

Primes of the Form n²+1 on the 6N Skeleton: the Wing Decomposition and the Character Spectrum of the Landau–Shanks Constant

Key Points

This study investigates the distribution of primes of the form n^2+1 and validates the Landau–Shanks constant as a measure of their density.
Enumerated all n ≤ 3×10^7 where n^2+1 is prime using a square-root sieve technique.
Analyzed the split among even integers in terms of 6N wing decomposition.
Assessed character spectrum of local factors associated with odd primes.
Identified 1,275,229 instances of n ≤ 3×10^7 for which n^2+1 is prime, with n^2+1 reaching 9×10^14.
Split in prime density on the 6N lattice showed a right:left ratio of 0.50015, indicating no bias.
Validated the Landau–Shanks constant C = 1.37281 against measured counts, with empirical values converging as N increases.

Abstract

Every constellation this series has measured so far — twin pairs, Goldbach sums, arithmetic progressions — is a linear translate of the prime set. We turn to the first nonlinear sequence: primes of the form n²+1, the subject of Landau's fourth problem. Using a square-root sieve on n (marking n ≡ ±r mod q where r² ≡ −1), we enumerate all n ≤ 3×10⁷ with n²+1 prime — 1, 275, 229 of them, with n²+1 as large as 9×10¹4 — and read the result through the 6N lattice. Three findings. (i) Wing decomposition: n must be even (odd n annihilates n²+1 into the even residue class), and among even n, 6 | n sends n²+1 to the right break-zone 6N+1 while n ≡ 2, 4 (mod 6) sends it to the left zone 6N−1. We measure the split as right: left = 0. 50015, exactly the 1: 2 of pure counting: the n²+1-prime density per channel is identical on the two wings, so the quadratic deformation induces no Chebyshev-type bias. (ii) Character spectrum: the local factor g (q) = (1 − ω (q) /q) / (1 − 1/q) splits the odd primes by the quadratic character of −1. Primes q ≡ 3 (mod 4) never divide n²+1 and enhance it by q/ (q−1) > 1; primes q ≡ 1 (mod 4) divide it on two residues and suppress it by (q−2) / (q−1) < 1. (iii) The product of these factors is the Landau–Shanks constant C = 1. 37281, and it controls the count: the measured Q (N) / ∫ dt/log (t²+1) equals 1. 37253, matching C to 0. 02%, with the empirical constant converging onto C as N grows. We verify the nonlinear (degree-two) Bateman–Horn heuristic to better than a part in 10³ at n²+1 ~ 10¹5. None of this is new as theory — ω (q), the constant C, and the asymptotic are due to Hardy–Littlewood, Bateman–Horn, and Shanks. What this paper adds is the 6N wing decomposition, the character-spectrum reading of the local factors, and a high-precision verification via the square-root sieve. We make no claim about the infinitude of such primes: Landau's problem is the backdrop, not a target. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input. Part XXXI of "Arithmetic Geodynamics on the 6N Skeleton. " Code and measured data: https: //github. com/Ruqing1963/6N-n2plus1

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