On a Problem of Diophantus

Long ago Diophantus of Alexandria [4] noted that the numbers 1/16, 33/16, 68/16, and 105/16 all have the property that the product of any two increased by 1 is the square of a rational number. Much later, Fermat [5] notes that the product of any two of 1, 3, 8, and 120 increased by 1 is the square of an integer. In 1969, Davenport and Baker [3] showed that if the integers 1, 3, 8 and c have this property then c must be 120. From this it follows that there does not exist an integer d dif f erent from 1, 3, 5, and 120 such that the five numbers 1, 3, 8, 120, and d have the same property. In 1977, Hoggatt and Bergum [7] noted that 1 = F2, 3 = F4, 8 = F6 and 120 = 4•2 •3 •5 = 4F1F3F5 where Fn is the nth Fibonacci number, and were led to guess that the numbers F2n, F2n+2, F2n+4, and 4F2n+1F2n+2F2n+3 possessed this same property for every n ≧ 1. Moreover, since 1, 2, and 5 have the property that the product of any two decreased by 1 is the square of an integer and 1 = F1, 2 = F3, and 5 = F5, they guessed that there must exist an integer y such that the product of any two of 1, 2, 5, and y decreased by 1 must be a perfect square. More generally, they guessed that for every n ≧0 there must exist an integer yn such that the product of any two of F2n+1, F2n+3, F2n+5, and yn decreased by 1 must be a perfect square. Their guesses were only partly correct; however, they were able to prove the following theorems.