Prime Powers Differing by One

Theorem

$8$ and $9$ are the only powers of prime numbers which differ by exactly $1$:

$2^3 + 1 = 3^2$

Proof

This is a direct consequence of 1 plus Perfect Power is not Prime Power except for 9.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $8$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $8$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$