# Definition:Ramanujan-Nagell Number

## Theorem

A Ramanujan-Nagell number is a positive integer of the form $2^m - 1$ which is also triangular.

## Also known as

A Ramanujan-Nagell number is also known as a triangular Mersenne number.

However, this uses the definition of a Mersenne number which is of the form $2^m - 1$ for all positive integers $m$, which differs from the preferred definition on $\mathsf{Pr} \infty \mathsf{fWiki}$ which limits $m$ to the set of prime numbers.

## Source of Name

This entry was named for Srinivasa Ramanujan and Trygve Nagell.

## Historical Note

Srinivasa Ramanujan conjectured in $1913$ that the equation $x^2 + 7 = 2^n$ had only $5$ integer solutions.

The same conjecture was made independently in $1943$ by Wilhelm Ljunggren.

The conjecture was proved in $1948$ by Trygve Nagell