Annika Conjecture: Perfect Squares from the Radicals of Consecutive Integers

This paper introduces the Annika Conjecture, concerning the arithmetic function F(n) = rad(n) + rad(n + 1), where rad(n) denotes the product of the distinct prime divisors of n. The conjecture states that there exist infinitely many positive integers n for which F(n) is a perfect square. The paper presents computational evidence obtained from an exhaustive search through 500 million, identifying 15,796 solutions in the investigated range. It also establishes several elementary structural properties of solutions, including the coprimality of the two radicals, the oddness of F(n), and the fact that n and n + 1 cannot both be squarefree when F(n) is a perfect square. Further computational observations concerning residue classes, squarefreeness, and perfect square members of consecutive pairs are discussed as potential directions for future investigation. The computational evidence and heuristic arguments presented do not constitute a proof of infinitude. The principal assertion remains a conjecture.