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Has the congruence $n\equiv\varphi(n) \pmod p$, with $p$ being an odd prime not dividing $n$, been examined before? Because it is easy to find solutions for $n$ with few primes in its decomposition ( …

T. Wright http://www.math.jhu.edu/~wright/carmichael.pdf proves that $lcm ((p_1-1),(p_2-1),...)$ over the prime factors of a carmichael number cannot be of the form $2^s$ How to prove that carmichael …

Has the following generalized version of the Lehmer's conjecture for the Euler totient function (original version: there are no composite solutions to the equation $n-1 \equiv 0 (\varphi(n))$) Find c …

Do we know something about numbers $x$ with $x-\varphi(x)=p^n$ for some $n$ and prime $p$ Are there any for each prime $p$?

Numerical analysis of the first several hundred n suggests the following inequality: $\varphi(3^n-2) \ge 2\cdot3^{n-1}$

Lehmer-Euler conjecture states, that there a no composite numbers $n$ satisfying $(n-1) \equiv 0(\varphi(n))$ Are there any results out there for $a(n-1) \equiv 0(\varphi(n))$, for integer $a$ esp. …

I tried to reproduce the results in a paper of Schemmel (where Schemmel's totients were introduced) $n$ consecutive numbers is an arithmetic sequence of numbers with common difference $n$ (e.g. 1,5, …

Let $p,q_i,i=1,2,..m$ be odd primes with integer $m>2$ Does this system of congruences have any solutions? $\prod_{i=1}^m(q_i-1)\equiv2(p^2)$ $\prod_{i=1}^mq_i\equiv2(p^3)$

Let $p,q_i, i \ge 1$ be primes, $m$ a positive integer. The equation $$ p.\prod_{i=1}^m(q_i-1)-(p-1).\prod_{i=1}^mq_i=2 $$ for $m=1$ has all twin primes $p,q_1=p+2$ as solution. Are there solutions …