Dem’janenko’s Theorem on Jeśmanowicz’ Conjecture Concerning Pythagorean Triples Revisited

Abstract

Let f, g be positive integers such that $$f>g$$\n , $$\\gcd (f,g)=1$$\n and $$f\\not \\equiv g \\pmod 2$$\n . In 1956, L.\xa0Jeśmanowicz conjectured that, for any positive integer n, the equation $$(*)$$\n $$\\left( (f^2-g^2)n\\right) ^x+(2fgn)^y=\\left( (f^2+g^2)n\\right) ^z$$\n has only the positive integer solution $$(x,y,z)=(2,2,2)$$\n . This is a far from solved problem in number theory. In 1965, V.A. Dem’janenko claimed that if $$n=1$$\n and $$f=g+1$$\n , then $$(*)$$\n has only the positive integer solution $$(x,y,z)=(2,2,2)$$\n . This result solves an important case of Jeśmanowicz’ conjecture. In this paper, on the one hand, using some properties on the representation of integers by binary quadratic primitive forms, we give a new and elementary proof for Dem’janenko’s result; on the other hand, using the Baker method and its p-adic form, we prove that if $$n>1$$\n , $$f=g+1$$\n and $$g=2^r$$\n , where r is a positive integer with $$r\\ge 80$$\n and $$r+1$$\n is an odd prime, then $$(*)$$\n has only the positive integer solution $$(x,y,z)=(2,2,2)$$\n . It can thus be seen that if $$n>1$$\n , then there are infinitely many pairs (f,\xa0g) with $$f=g+1$$\n which make Jeśmanowicz’ conjecture is true.