Harman sieve and Lean-verified certificate for the \(n^{1.323}\) bound on the greatest prime factor of \(n^2+1\).
Prime quintuplets and the sextuplet boundary for Q(n)=n^47-(n-1)^47 (Titan Project, Part III). Exact Bateman-Horn singular series via Dirichlet L-functions mod 47; parameter-free prediction of 7 quintuplets / 0 sextuplets up to 2e11; first sextuplet expected near N*~1e13. Paper, data, scripts. v2 DOI 10.5281/zenodo.23178170
Primes p ≤ N with N − p an almost prime sum of two squares (Ω ≤ 9): paper, semi-linear vector sieve with Richert weights and switching, and an interval-arithmetic certification script.
Manuscript, certified computations and data for "Recovering the level of distribution x^{1/2} for the orders of y² = x³ − x modulo p": the Iwaniec–Jiménez Urroz P₂ theorem holds as stated, and Xie's prime-power bounds improve to P₃ and ⌊2.05(ℓ−1)+1.2⌋.
A very fast Prime Sieve
Bateman–Horn census of prime constellations of Q(n)=n^q−(n−1)^q: 2,393 Q47 quadruplets below 8×10^11 (BPSW-verified), seven-exponent recount, ECPP-certified 1000-digit quadruplets, and a 200,000-digit Q47 PRP. Code, data, certificates, paper v2.
Sums of two squares with few prime factors via theta series of quaternary forms: Ω(n(N−n)) ≤ 9 and max Ω ≤ 6 for n, N−n both sums of two squares, from a level of distribution 2/5 for r(n)r(N−n). Manuscript, interval-arithmetic sieve certification, numerical experiments (v2.0).
N = n₁ + n₂ with n₁, n₂ sums of two squares (or values of any class number one form) and Ω(n₁n₂) ≤ 11, max Ω(nᵢ) ≤ 8: paper, linear vector sieve with switching and Richert weights, interval-arithmetic certification.
Finite local sieves for shifted even squares, with exact CRT densities and prime-square extensions.
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