On Arithmetic Progressions and a Proof of the Nonexistence of Magic Squares of Squares
Oscar Hill
TL;DR
The paper tackles the long-standing question of whether a 3×3 magic square with distinct square entries can exist. It develops a framework based on pairs of consecutive odd arithmetic progressions with equal sums, introducing offsets and the ratio κ to relate AP components, and connects these structures to constraints arising in a putative 3×3 square. The main result is a contradiction: the equal-sum AP-pair framework forces $P_1^{n_2}=P_2^{n_2}$ and $P_1^1=P_2^1=P_3^1$, implying $P_1=P_2=P_3$, which violates the distinctness requirement. Consequently, there is no 3×3 magic square consisting solely of square integers, and the AP-pair approach provides a clear route to nonexistence results for small-order magic squares.
Abstract
We explore some of the properties of consecutive arithmetic progressions of odd numbers with equal sums, particularly their offsets and said sums, before using them to prove that no $3\times3$ magic squares of distinct square integers exist.