  We make an analytical proof for Lehmer's totient conjecture. Lehmer
conjectured that there is no solution for the congruence equation $n-1\equiv
0~(mod~\phi(n))$ with composite integers,$n$, where $\phi(n)$ denotes Euler's
totient function. He also showed that if the equation has any composite
solutions, $n$ must be odd, square-free, and divisible by at least 7 primes.
Several people have obtained conditions on values ,$n$, and number of
square-free primes constructing $n$ if the equation can have composite
solutions. Using Mertens' theorems, we show that it is impossible that the
equation can have any composite solution and implies that the conjecture should
be true for all the positively composite numbers.
