不定方程φ(abc)=2φ(a)φ(b)+6φ(c)的可解性

The Solvability of Diophantine Equationφ(abc)=2φ(a)φ(b)+6φ(c)

在gcd(a,b)=p,gcd(a,c)=1,gcd(b,c)=1(p为素数)的条件下研究不定方程Φ(abc)=2Φ(a)Φ(b)+6Φ(c)的可解性问题,利用初等方法求出了方程在p=1,2,3,5,7时所有解,共61组,且该方法具有一定普适性,可以推广到类似的方程.

The solvability of the Diophantine equation,under the condition of gcd(a,b)=p,gcd(a,c)=1,gcd(b,c)=1(p is prime number).All the solutions of the equation at p=1,2,3,5,7 are obtained by using the preliminary mathematical methods,which is 61 groups in total,and the method has certain universal applicability,which can be extended to similar equations.