  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.
