Five Ramanujan-Nagell Numbers

Theorem
There exist exactly $5$ Ramanujan-Nagell numbers: positive integers of the form $2^m - 1$ which are triangular:
 * $0, 1, 3, 15, 4095$

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

Let:
 * $n = m - 3$
 * $x = 2 r + 1$

and it can be seen that $(1)$ is equivalent to:
 * $x^2 + 7 = 2^n$

From Solutions of Ramanujan-Nagell Equation:
 * $x = 1, 3, 5, 11, 181$

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:
 * $0, 1, 3, 15, 4095$

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