An Infinitude of Primes of the Form $b^2 + 1$

TL;DR

This paper proves there are infinitely many primes of the form b^2 + 1 by linking composite cases to sums of two squares and using Gauss's results to show such composites cannot be infinite.

Contribution

It introduces a novel connection between composites of the form 4u^2 + 1 and sums of two squares related to consecutive integers, establishing the infinitude of primes of the form b^2 + 1.

Findings

Infinitely many primes of the form b^2 + 1 exist.

Composite numbers of the form 4u^2 + 1 are linked to sums of two squares.

Gauss's results are used to prove the impossibility of infinite such composites.

Abstract

If b^2 + 1 is prime then b must be even, hence we examine the form 4u^2 + 1. Rather than study primes of this form we study composites where the main theorem of this paper establishes that if 4u^2 + 1 is composite, then u belongs to a set whose elements are those u such that u^2 + t^2 = n(n + 1), where t has a upper bound determined by the value of u. This connects the composites of the form 4u^2 + 1 with numbers expressible as the sum of two squares equal to the product of two consecutive integers. A result obtained by Gauss concerning the average number of representations of a number as the sum of two squares is then used to show that an infinite sequence of u for which u^2 + t^2 = n(n + 1) is impossible. This entails the impossibility of an infinite sequence of composites, and hence an infinitude of primes of the form b^2 + 1.