109

Number

$109$ (one hundred and nine) is:

The $5$th prime number after $53$, $71$, $103$, $107$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$.

The $8$th of $11$ primes of the form $2 x^2 + 11$:

$2 \times 7^2 + 11 = 109$ (Previous ... Next)

The larger of the $10$th pair of twin primes, with $107$

The $10$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$

The $22$nd happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$:

$109 \to 1^2 + 0^2 + 9^2 = 1 + 0 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$