与Euler函数φ(n)有关的非线性方程的正整数解

Positive integer solution of nonlinear equations related to Euler function φ(n )

在Euler函数φ(n)的性质的基础上,利用整数分解的方法证明了对任意的正整数m,n,非线性方程φ(mn)=aφ(m)+bφ(n)+c^2(a,b,c为勾股数且gcd(a,b,c)=1)当(a,b,c)=(3,4,5),(5,12,13),(7,24,25)时无正整数解,并证明了当a,b为任意的一奇一偶,c为任意的奇数,且满足a^2+b^2=c^2,gcd(a,b)=1,2|b时,方程无正整数解.

In this paper, basing on the properties of Euler function φ（n）, it is proved that for any positive integer m, n the nonlinear equation φ（mn） = aφ（m） ＋ bφ（n） ＋ c^2（a, b, c as the Pythagorean number and gcd（a, b, c） = 1）has no positive integer solution when（a, b, c） =（3, 4, 5）,（5, 12, 13）,（7, 24, 25） by integer factorization method.When a, b are any odd even pairs, c is any odd number, and a^2＋ b^2= c^2, gcd（a, b） = 1, 2 ｜ b are satisfied, it is proved that there is no positive integer solution.