In solving Diophantine problems, one must guard against falling into certain formal traps. The identity{}

might lead one to think that, because all numbers of the form are squares, any number of say, the form could never be a perfect square because it is not an algebraic square. But, numbers are more versatile than that. Try in the expression .

It was once proposed as a theorem that “the product of four consecutive terms of an AP of integers plus the fourth power of the common difference is always a perfect square but never a perfect fourth power.” If a is the first term of the four and b the common difference, we have which works out to be the same thing as .

The truth of the first assertion of the theorem is now evident, but we have already given a counter example showing the second assertion to be incorrect.

Ref: Excursions in Number Theory, C. Stanley Ogilvy and John T. Anderson,

–*Nalin Pithwa.*