2013/11/21 by Konstantine Zelator, Zelator, Konstantine
Mathematics · #History and Theory of Mathematics #Analytic Number Theory Research #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1311.6484
In a paper published by this author in www.academia.edu(see reference[3]), it\nwas established that there exist no three positive integers which are\nconsecutive terms of an arithmetic progression; and whose sum of squares is a\nperfect or integer square. In that paper, we made use of the 3-parameter\nformulas which describe the entire set of positive integer solutions of the\n4-variable equation, x2+y2+z2= t2 (See reference [1]) In this work, we\noffer an alternative proof to the above result; a proof that uses only powers\nof 3 divisibility arguments. This is done in Theorem1, Section2. After that, in\nProposition3(Section4) we use the Quadratic Reciprocity Law for odd primes, to\nestablish that if p is a prime congruent to 5 or 7 modulo12; then 3 is\nquadratic non-residue of p. This then, plays a key role in proving Theorem 2,\nwhich postulates that if p is an odd prime congruent to 5 or 2 mod12; then, the\nsum of the squares any p natural numbers which are consecutive terms of an\narithmetic progression; is a non-perfect square. Theorem3 is an immediate\ncorollary of Theorem2 : for primes p congruent to 5 or 7 mod12; the sum of the\nsquares of p natural numbers, consecutive terms of an arithmetic progression;\ncannot be a perfect square.\n