关于商高数的Jesmanowicz猜想

The Jesmanowicz conjecture of Pythagorean triples

本研究主要利用简单同余、二次剩余、κ次剩余、四次剩余特征理论及因式分解法,对关于不定方程a^(x)+b^(y)=c^(z)的Jesmanowicz猜想的一类特殊情形进行证明,并得到如下结论:定理对于商高数组a=n^(2)-4,b=4n,c=n^(2)+4,2×n,当n+2含有素因子p■-1(mod 16)时,Jesmanowicz猜想成立.特别地,有推论对于上述商高数组,当n■-1(mod 16)时,Jesmanowicz猜想成立.

We proved that the conjecture of Jesmanowicz concerning Pythagorean triples for the diophantine equation a^(x)+b^(y)=c^(z) held true in a special cases.Based on elementary congruence,quadratic residue,bi-quadratic residue characters and factorization method.Theorem For the pythagorean numbers a=n^(2)-4,b=4n,c=n^(2)+4,2×n,the conjecture of Jesmanowicz holds when n+2 exists a prime factor p such that p■1(mod 16).In particular,we have that Corollary For the pythagorean numbers,the conjecture of Jesmanowicz holds when n such that n■1(mod 16).