Solutions of Ramanujan-Nagell Equation

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Theorem

Integer solutions to the Ramanujan-Nagell equation:

$x^2 + 7 = 2^n$

exist for only $5$ values of $n$:

$3, 4, 5, 7, 15$

This sequence is A060728 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The corresponding values of $x$ are:

$1, 3, 5, 11, 181$

This sequence is A038198 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

By direct implementation:

\(\text {(1)}: \quad\) \(\ds 1^2 + 7\) \(=\) \(\ds 1 + 7\)
\(\ds \) \(=\) \(\ds 8\)
\(\ds \) \(=\) \(\ds 2^3\)
\(\text {(2)}: \quad\) \(\ds 3^2 + 7\) \(=\) \(\ds 9 + 7\)
\(\ds \) \(=\) \(\ds 16\)
\(\ds \) \(=\) \(\ds 2^4\)
\(\text {(3)}: \quad\) \(\ds 5^2 + 7\) \(=\) \(\ds 25 + 7\)
\(\ds \) \(=\) \(\ds 32\)
\(\ds \) \(=\) \(\ds 2^5\)
\(\text {(4)}: \quad\) \(\ds 11^2 + 7\) \(=\) \(\ds 121 + 7\)
\(\ds \) \(=\) \(\ds 128\)
\(\ds \) \(=\) \(\ds 2^7\)
\(\text {(5)}: \quad\) \(\ds 181^2 + 7\) \(=\) \(\ds 32 \, 761 + 7\)
\(\ds \) \(=\) \(\ds 32 \, 768\)
\(\ds \) \(=\) \(\ds 2^{15}\)


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Also see


Sources

Beware of the terminological mistake: he omits to point out that the equation is Diophantine.
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