For a non-negative integer n, let g(n) be the smallest integer s≥ n for which there are distinct integers a_1,…,a_t∈[n,s], with a_1=n, such that ∏_i=1^t a_i is a square. Kagey–Rajesh's conjecture. For all n∈N, g(n)=2n if and only if n>3 is prime, n=0, or n=6. The authors proved the “if” direction; this paper proves the “only if” direction, so the conjecture is solved.

Kagey–Rajesh characterization of the equality g(n)=2n

In 2024, Peter Kagey and Krishna Rajesh formulated the characterization of when the smallest square-producing interval reaches $2n$ . Their work established one direction and left the converse as a conjecture.

Known results

Kagey and Rajesh (2024) proved the “if” direction and gave structural results, an algorithm, and bounds for $g(n)$ .
Their paper stated the converse characterization as Conjecture 37.

April 27, 2025 claimed completion

The preprint On a Generalisation of a Function of Ron Graham’s states that its Theorem 3.4 proves the “only if” direction, giving $g_2(n)=2n$ exactly when $n>3$ is prime, $n=0$ , or $n=6$ . This would solve the stated Kagey–Rajesh conjecture, but the retrieved sources provide no independent verification, gap report, or publication confirmation.

Current status (as of September 2026): The conjecture is claimed solved by the April 2025 preprint, but that claim remains unverified in the retrieved record.

Kagey–Rajesh characterization of the equality g(n)=2n

Primary source

Progress summary

Solutions 0