Sierpinski's Hypothesis H1

TL;DR

The paper addresses Sierpiński's Hypothesis $H_1$, which posits that each row of the $n\times n$ Sierpiński matrix $S_n$ contains a prime. It combines maximal prime-gap data, the pigeonhole principle, and bounds on the first Chebyshev function to prove unconditional validity of $H_1$ for all $n$ up to $N_{83}=\left\lfloor\sqrt{p^*_{83}}\right\rfloor=4{,}553{,}432{,}387$, and derives additional partial results for larger $n$, including a fractional guarantee that at least a quarter of the rows contain primes and a bound ensuring the first $131{,}294$ rows do so. The work extends to contiguous-row considerations and provides explicit lower bounds on primes per row (via $i_{min}$ and maximal gaps) and Chebyshev-based bounds to guarantee primes in substantial portions of the matrix. These results sharpen our understanding of prime distribution in structured arrays and tie the problem to classical conjectures, while also outlining clear directions for improvement via new prime-gap data and stronger analytic bounds. The methods offer rigorous, scalable guarantees that could be refined further as more maximal prime gaps and sharper $\vartheta$-bounds become available.

Abstract

Sierpinski's Hypothesis H1, formulated in 1958, is the conjecture that (provided $n\geq 2$), when the first $n^2$ counting numbers, $1, 2,3,\dots n^2$, are arranged in a square, then each row contains at least one prime. This conjecture is particularly interesting in that it subsumes and is stronger than both the Oppermann and Legrendre conjectures. Herein I shall verify Sierpinski's Hypothesis H1 for (at least) the first $n \leq \hbox{4 553 432 387} \approx 4.5 \hbox{ billion}$ of these Sierpinski matrices. I shall also demonstrate some partial but more general results. For example: Even for arbitrary $n\geq \hbox{4 553 432 388}$ at least one quarter of the rows of the $n$th Sierpinski matrix contain at least one prime. Furthermore, even for arbitrary $n\geq \hbox{4 553 432 388}$ at least the first $\hbox{131 294}$ rows of the $n$th Sierpinski matrix always contain at least one prime. These and related results are obtained largely by using the locations and values of the known maximal prime gaps, the pigeonhole principle, and some recent bounds on the first Chebyshev function.