Sierpinski's Hypothesis H1

TL;DR

This paper verifies Sierpinski's Hypothesis H1 for the first approximately 4.5 billion cases and provides partial results for larger n, using prime gap data, the pigeonhole principle, and bounds on the Chebyshev function.

Contribution

The paper confirms Sierpinski's Hypothesis H1 up to 4.5 billion and establishes partial results for larger matrices, advancing understanding of prime distribution in structured arrangements.

Findings

Verified H1 for n up to 4,553,432,387.

At least 25% of rows contain a prime for larger n.

First 131,294 rows always contain a prime for larger n.

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.