, , and are integers such that and .
How many ordered sets of integers satisfy
where is a prime number?
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C l a i m : If a , b , c , d are positive integers satisfying a b = c d , then a n + b n + c n + d n is composite for any natural n . P r o o f : Now, a b = c d , then there exists co-prime integers m , k such that d a = b c = k m . So for any integers p , q , the solution of a b = c d is ( a , b , c , d ) = ( m p , k q , m q , k p ) . So a n + b n + c n + d n is equal to ( m n + k n ) ( p n + q n ) , on factorising . Indeed this is composite and each term is greater than 1 .