# Equations involving the Euler's totient function and Mersenne primes

In this post we denote the Euler's totient function as $\varphi(n)$, first we show a claim related to Mersenne primes, see for example this Wikipedia and secondly we are going to ask a related conjecture.

We denote as $m=2^p-1$ a Mersenne prime, and we assume that there exists $k\geq 1$ an integer satisfying that $M=2^{p+2k}-1$, whenever $m<M$ are distinct Mersenne primes. With these definitions it is possible to prove the following claim.

Claim. If $m$ is a Mersenne prime and $1\leq k$ is an integer such that $2^{2k}m+2^{2k}-1$ is also a Mersenne prime, then the pair $(m,k)$ is a solution of the equation

$$(m^2-1)\left((2^{2k}m+2^{2k}-1)^2-1\right)=16\varphi\left(m(2^{2k}m+2^{2k}-1)\right)\varphi(\frac{m+1}{2})\varphi\left((m+1)2^{2k-1}\right).$$

Question. Prove or refute, showing a counterexample, the following conjecture:

Let $1\leq z$ an integer, and $1\leq\kappa$ also integer satisfying $$(z^2-1)\left((2^{2\kappa}z+2^{2\kappa}-1)^2-1\right)=16\varphi\left(z(2^{2\kappa}z+2^{2\kappa}-1)\right)\varphi(\frac{z+1}{2})\varphi\left((z+1)2^{2\kappa-1}\right).$$ Then $z$ is a Mersenne prime, and $2^{2\kappa}z+2^{2\kappa}-1$ is also a Mersenne prime.

Many thanks.

• How far have you checked your conjecture? – Jose Arnaldo Bebita-Dris May 10 '18 at 3:55
• Many thanks for your attention and help @JoseArnaldoBebitaDris , for example using Sage Cell Server I can to check it for $1\leq z\leq 10^4$ and $1\leq \kappa\leq 20$ (I accept of course that this $\kappa$ is very small as computational evidence). The code that I've used is this line written in Pari-GP for (z = 1, 10000,for (k = 1, 20, if (z%2==1&&(z^2-1)*((2^(2*k)*z+2^(2*k)-1)^2-1)==16*eulerphi((z+1)/2)*eulerphi((z+1)*2^(2*k-1))*eulerphi(z*(2^(2*k)*z+2^(2*k)-1)), print(z)))) – user243301 May 10 '18 at 7:18
• I did run your Pari-GP code using Sage Cell Server, @user243301. However, they may have been some issues, as the green window was still there, and there was no output. I also tried using a different browser, still no go. – Jose Arnaldo Bebita-Dris May 13 '18 at 9:39
• @JoseArnaldoBebitaDris I did run it now, and it works, try to run the same code but put different upper limits for $z$ and/or $k$. That is an upper limit less than 10000 for $z$ or an upper limit less than 20 for $k$. Thus in my compute previous code works, on the other hand when the upper limits are very big Sage Cell Server send you the warning PARI/GP interpreter crashed -- automatically restarting in the green window. – user243301 May 13 '18 at 9:54
• Which language do I select? GP? – Jose Arnaldo Bebita-Dris May 13 '18 at 9:59