The sum of the squares of p positive integers which are consecutive terms of an arithmetic progression: Always a non-perfect square when p(a prime)=3 or p is congruent to 5 or 7 modulo 12

Abstract

In a paper published by this author in www.academia.edu(see reference[3]), it was established that there exist no three positive integers which are consecutive terms of an arithmetic progression; and whose sum of squares is a perfect or integer square. In that paper, we made use of the 3-parameter formulas which describe the entire set of positive integer solutions of the 4-variable equation, $x^2+y^2+z^2= t^2$ (See reference [1]) In this work, we offer an alternative proof to the above result; a proof that uses only powers of 3 divisibility arguments. This is done in Theorem1, Section2. After that, in Proposition3(Section4) we use the Quadratic Reciprocity Law for odd primes, to establish that if p is a prime congruent to 5 or 7 modulo12; then 3 is quadratic non-residue of p. This then, plays a key role in proving Theorem 2, which postulates that if p is an odd prime congruent to 5 or 2 mod12; then, the sum of the squares any p natural numbers which are consecutive terms of an arithmetic progression; is a non-perfect square. Theorem3 is an immediate corollary of Theorem2 : for primes p congruent to 5 or 7 mod12; the sum of the squares of p natural numbers, consecutive terms of an arithmetic progression; cannot be a perfect square.