Odd Squares 7 Less than Nearest Power of 2

Theorem

There exist exactly $3$ odd squares which are $7$ less than the nearest power of $2$:

$5^2 = 25 = 2^5 - 7$

$11^2 = 121 = 2^7 - 7$

$181^2 = 32 \, 761 = 2^{15} - 7$

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

Note that this sequence includes $1$ and $3$, being all the squares which are $7$ less than a power of $2$.

However, for $1$ and $3$, those powers ($8$ and $16$ respectively) are not the nearest power of $2$ ($1$ and $4$ respectively).

Proof

From Solutions of Ramanujan-Nagell Equation, the only solutions to the equation:

$x^2 + 7 = 2^n$

are $\tuple {x, n} =$:

$\tuple {1, 3}, \tuple {3, 4}, \tuple {5, 5}, \tuple {11, 7}, \tuple {181, 15}$

so no more solutions exist.

$\blacksquare$

Sources

• 1992: F. Beukers: On the generalized Ramanujan-Nagell equation I (Acta Arith. Vol. 38: pp. 389 – 410)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$