Powerful Prime Numbers

Number Theory Level pending

Let $M$ be a $2015 \times 2015$ matrix with elements defined as

$M_{p,q} = 2^{(p+1)\times(q+1)}-1$ , where $p,q$ are integers with $0<p,q \leq 2015$ .

How many elements of $M$ are prime.

2015 1 0 None of the Others 621

1 solution

Janardhanan Sivaramakrishnan
Jan 18, 2015

$M_{p,q}=(2^{(p+1) \times (q+1)}-1)=(2^{(q+1)}-1)\times(\sum_{i=0}^{p} 2^{i \times (q+1)})$

Since both $p,q \geq 1$ , both the terms in the RHS are greater than 1. Hence, each $M_{p,q}$ is composite. The required number is thus $\boxed{0}$