关于一类勾股数的Je?manowicz猜想

A Conjecture of Je?manowicz Concerning Pythagorean Triples

设k,l,m1,m2是正整数,p,q为素数,满足p^(k)=2^(m1)−3^(m2),q^(l)=2^(m1)+3^(m2),且2■m2或2|m1,2|m2.本文证明了对任意正整数n,丢番图方程(q^(2l)−p^(2k)/2n)^(x)+(p^(k)q^(l)n)^(y)=(q^(2l)+p^(2k)/2n)^(z)除x=y=z=2外没有其他的正整数解.从而说明Jesmanowicz猜想在该类情形下成立.

Let k,l,m1,m2 be positive integers,p,q be primes such that p^(k)=2^(m1)−3^(m2),q^(l)=2^(m1)+3^(m2),and 2■m2 or 2|m1,2|m2.In this paper,we show that for any positive integer n,the Diophantine equation (q^(2l)−p^(2k)/2n)^(x)+(p^(k)q^(l)n)^(y)=(q^(2l)+p^(2k)/2n)^(z) has no solution in positive integers other than x=y=z=2.The result is still the confirmation of Jesmanowicz's conjecture.