Tootfinder

A Constructive Heuristic Sieve for the Twin Prime Problem
Yuhang Shi
https://arxiv.org/abs/2507.03107 https://arxiv.org/pdf/2507.03107

A Constructive Heuristic Sieve for the Twin Prime Problem
The quantitative distribution of twin primes remains a central open problem in number theory. This paper develops a heuristic model grounded in the principles of sieve theory, with the goal of constructing an analytical approximation for the twin prime constant from first principles. The core of this method, which we term ``$f(t; z)$ function analysis'', involves representing the sieve's density product as a ratio of infinite series involving $f(t;z)$, the elementary symmetric polynomials of pr…

This https://arxiv.org/abs/2405.05727 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On Chen's theorem, Goldbach's conjecture and almost prime twins II
Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,2}(N)$ as the number of primes $p$ that such that $N - p$ has at most 2 prime factors. In this paper, we show that $D_{1,2}(N) \geqslant 1.9728 \frac{C(N) N}{(\log N)^2}$, which is rather near to the asymptotic constant $2$ in Hardy--Littlewood conjecture for Goldbach's conjecture. We also get similar results on twin prime problem and additive representations of integers. The proof comb…

Replaced article(s) found for math.GM. https://arxiv.org/list/math.GM/new
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- On The Infinitude of the Twin Primes
Berndt Gensel
https://

Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728
Ben Savoie
https://arxiv.org/abs/2506.17605 https://

Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728
We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ of the form $y^2 = x^3 - γ^2 x$, where $γ\in \mathbb{Z}[i]$, with rank 2. In addition, we prove that if $p$ and $q$ are rational twin primes with $p \equiv 5 \bmod 8$, then $y^2 = x^3 + p q x$ has rank 2 over $\mathbb{Q}(i)$. Lastly, we prove that if $p$ is a rational prime of the form $p = a^2 + b^4$ (of which there are infinitely many) and $p \not\equiv 1 \bmod 16$, then $y^2 = x^3 - p x$ has rank 2 over $\mathbb…