关于一类商高数的Jeśmanowicz猜想

A Conjecture of Jeśmanowicz Concerning Pythagorean Triples

设k,l,m1,m2是正整数,p,q是奇素数满足p^(k)=2^(m1)-a^(m2),q^(l)=2^(m1)+a^(m2),这里a≡3(mod8)或a≡5(mod8)为素数.利用因式分解、同余和柯召方法等基本方法,证明了指数丢番图方程(q^(2l)-p^(2k)/2 n)x+(p^(k)q^(l)n)y=(q^(2l)+p^(2k)/2 n)z仅有正整数解(x,y,z)=(2,2,2).证明了Jeśmanowicz猜想对商高数q^(2l)-p^(2k)2 n,p^(k)q^(l)n,q^(2l)+p^(2k)2 n成立,从而改进文献的工作,推广文献工作.

Let l,k,m1,m2 be positive integers,p,q be odd primes satisfying p^(k)=2^(m1)-a^(m2),q^(l)=2^(m1)+a^(m2),where a≡3(mod8)or a≡5(mod8)are prime numbers.By using only the elementary methods of factorization,congruence methods and Ke Zhao methods,the exponential Diophantine equathion(q^(2l)-p^(2k)/2 n)x+(p^(k)q^(l)n)y=(q^(2l)+p^(2k)/2 n)z had only the positive integer solution(x,y,z)=(2,2,2).We proved the conjecture for Pythagorean number q^(2l)-p^(2k)/2 n,p^(k)q^(l)n,q^(2l)+p^(2k)/2 n.Thus,the work of literature was improved.