"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way." (G. H. Hardy, A Mathematician's Apology, 1940)
317 has a representation as a sum of two squares: 317 = 112 + 142.
317 is the hypotenuse of a primitive Pythagorean triple: 3172 = 752 + 3082.
The sum of the squares of the digits of the prime 317 is 59, another prime. Note that all odd digits are present.
317, 31, and 3 are prime numbers and so are 17 and 7.
Former pitching ace Walter "Big Train" Johnson once hurled a coin 317 feet across the Rappahannock River in Virginia.
Source: Prime Curios!