Research on Euler Totient function equation k(n)=n-1

Abstract

Let denote Eulers Totient function. There are some properties about (n), when n is a prime or n=p_1^(r_1 )p_k^(r^k ). The Eulers function equation, k(n)=n-1(1), where k is a positive integer, and n is a composite number, is called Lehmers conjecture. Lehmer mentioned a series of properties of n that satisfy the equation in his own thesis and provided some proof. Afterwards, Ke Zhao and Sun Qi conducted further research. In previous studies, this conjecture was considered correct, but it is difficult to prove it. The case k=2 has been discussed and proved that when k=2 and n=p_1 p_2,... p_i are different prime numbers. Also, some properties of the composite numbers that satisfy the equation have also been proven. Some conclusions can be proven, by using elementary number theory methods. Using these conclusions, we can conclue that when k=2, the solution of (1) is at least the product of 12 odd prime numbers.