Perfect Squares from the Radicals of Consecutive Integers

This paper investigates the conjecture that there exist infinitely many positive integers n for which the sum of the radicals of two consecutive integers is a perfect square: rad(n) + rad(n + 1) = m^2. An exhaustive computational search through n <= 5 x 10^8 found 15,796 solutions. The paper establishes several structural properties of the solutions, including the coprimality of the two radicals, the oddness of the radical sum, and the fact that n and n + 1 cannot both be squarefree in a solution. A further structural result concerning the 2-adic valuation of the even member of a solution pair is proved. The paper also presents computational observations and heuristic arguments concerning the growth of the number of solutions. The computational evidence is consistent with the conjecture, but no proof of infinitude is claimed.