Five Ramanujan-Nagell Numbers

Theorem

There exist exactly $5$ Ramanujan-Nagell numbers: positive integers of the form $2^m - 1$ which are triangular:

$0, 1, 3, 15, 4095$

Consider the numbers of the form $2^m - 1$ which are triangular:

$\ds 2^m - 1$

$=$

$\ds \frac {r \paren {r + 1} } 2$

$\ds \leadstoandfrom \ \ $

$\ds 8 \paren {2^m - 1}$

$=$

$\ds 4 r \paren {r + 1}$

$\ds \leadstoandfrom \ \ $

$\ds 2^{m + 3} - 8$

$=$

$\ds 4 r^2 + 4 r$

$\text {(1)}: \quad$

$\ds \leadstoandfrom \ \ $

$\ds 2^{m + 3} - 7$

$=$

$\ds 4 r^2 + 4 r + 1$

$\ds $

$=$

$\ds \paren {2 r + 1}^2$

Let:

$n = m - 3$

$x = 2 r + 1$

and it can be seen that $(1)$ is equivalent to:

$x^2 + 7 = 2^n$

$x = 1, 3, 5, 11, 181$

Setting $r = \dfrac {x - 1} 2$ it is seen that the corresponding triangular numbers are:

$\dfrac {\paren {x - 1} \paren {x + 1} } 8$

Thus the corresponding Ramanujan-Nagell numbers are:

$0, 1, 3, 15, 4095$

$\blacksquare$