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

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

2010-07-11

arxiv.org/abs/1007.1786v2

Mathematics

General Mathematics

Due to comments from other mathematicians, I renamed the variable used in Def. 2.2 and expanded the proof given in section 2.

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

An Infinitude of Primes of the Form $b^2 + 1$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with An Infinitude of Primes of the Form $b^2 + 1$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An Infinitude of Primes of the Form $b^2 + 1$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-563510