Solutions of Ramanujan-Nagell Equation

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

Five Ramanujan-Nagell Numbers

Sources

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$

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