Fermat's Enigma

by Simon Singh

Fermat noticed that 26 is sandwiched between 25 and 27, one of which is a square number (25 = 52 = 5 × 5) and the other is a cube number (27 = 33 = 3 × 3 × 3). He searched for other numbers sandwiched between a square and a cube but failed to find any, and suspected that 26 might be unique. After days of strenuous effort he managed to construct an elaborate argument that proved without any doubt that 26 is indeed the only number between a square and a cube. His step-by-step logical proof established that no other numbers could fulfill this criterion. Fermat announced this unique property of 26 to the mathematical community, and then challenged them to prove that this was the case. He openly admitted that he himself had a proof; the question was, however, did others have the ingenuity to match it?