257

257 is a prime number. It is the second largest known Fermat prime.

257 is 100000001 in base 2 (binary).

257 has a representation as a sum of two squares: 257 = 1² + 16².

257 is the hypotenuse of a primitive Pythagorean triple: 257² = 32² + 255².

There are 257 topologically distinct polyhedral graphs of eight nodes.

257 is the smallest three-digit prime with distinct prime digits.

257 Silesia is an asteroid discovered in 1886.

Source: Number Gossip

17

17 is a prime number. It is the third Fermat prime: 17 = 2^{2^2} + 1.

17 has a representation as a sum of two squares: 17 = 1² + 4².

17 is the smallest number with four representations as a sum of three primes: 17 = 2 + 2 + 13 = 3 + 3 + 11 = 3 + 7 + 7 = 5 + 5 + 7.

17 and 19 form a twin prime pair.

17 is the hypotenuse of a primitive Pythagorean triple: 17² = 8² + 15².

17 is the first sum of two distinct fourth powers: 17 = 1⁴ + 2⁴.

There are 17 ways to express 17 as the sum of one or more primes; 17 is the only integer that is equal to the number of prime partitions of itself.

Carl Friedrich Gauss proved at the age of 18 that one can construct a regular polygon with a prime number of sides with the use of only straight edge and compasses only if the number of sides is of the form 2²^n. Hence, one can construct a regular 17-gon with rule and compasses only.

Source: Number Gossip