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- W2161648739 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." @default.
- W2161648739 created "2016-06-24" @default.
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- W2161648739 date "2010-07-11" @default.
- W2161648739 modified "2023-09-27" @default.
- W2161648739 title "An Infinitude of Primes of the Form $b^2 + 1$" @default.
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