As we can ask if the set of Mersenne primes is infinite we can ask also if the set of Lehmer-Ramanujan primes is infinite. Of course in both cases, we have necessary conditions which use the self similarity of primality i.e if 2 ^p -1is prime then p is prime and if τ(p ^q-1 ) is prime then p and q are odd primes. With those elements we do not have some sufficient conditions. Ramanujan has found two formulas which have been proved by Mordell i.e. τ(mn)=τ(m)τ(n) for m and n coprimes τ(p ⁿ )=τ(p)τ(p ^n-1 )-p ¹¹ τ(p ^n-2 ) with p prime. Let’s consider the L-R set, the set of all L-R prime numbers.

L-R set:= {τ (p ^q-1 ) ∈ Π}

Our answer to the initial question is the following conjecture.

L-R conjecture │{ τ(p ^q-1 ) ∈ Π}│˃ +∞