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In this work we show that based on a conjecture for the pair correlation ofintegers representable as sums of two squares, which was first suggested byConnors and Keating and reformulated here, the second moment of thedistribution of the number of representable integers in short intervals isconsistent with a Poissonian distribution, where "short" means of lengthcomparable to the mean spacing between sums of two squares. In addition wepresent a method for producing such conjectures through calculations in primepower residue rings and describe how these conjectures, as well as the abovestated result, may by generalized to other binary quadratic forms. Whileproducing these pair correlation conjectures we arrive at a surprising resultregarding Mertens' formula for primes in arithmetic progressions, and in orderto test the validity of the conjectures, we present numericalz computationswhich support our approach.

Sum of Two Squares - Pair Correlation and Distribution in Short Intervals