Problem of the Fortnight #12
Pythagorean
Triplets and Quadruplets!
It is a well-known fact that
there are infinitely many solutions to the equation
in positive integers a,
b and c having greatest common divisor equal to 1. Moreover, it is known that for any such
solution, c is an odd integer; and
thus 
is also an odd integer.
Prove, possibly using the
fact just mentioned, that the equation
has infinitely many solutions
in positive integers a, b, c and d having greatest common divisor equal to 1 and which satisfy the
additional condition that c and d are consecutive integers.
Solutions are due by