The number of ones that form the fourth repunit prime. It was first proven prime by Hugh C. Williams in 1977. [Dobb]
Romanian mathematician Dimitrie Pompeiu (1873-1954) posed the following puzzle: ABC398246 is a nine-digit number exactly divisible by 317, whose first three digits (A,B,C) are unknown. What are the digits A, B, and C?
"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)
The sum of the squares of the digits of the prime 317 is 59, another prime. Note that all odd digits are present. [Trotter]
317 = (-3)3 + 13 + 73. [Dobb]
The giant statue of Buddha in the lamasery of Yung-Ling, China, doubles as an object of veneration and a library containing 317 volumes of Tibetan classics. [Dobb]
Yasutoshi Kohmoto currently holds the record for the largest known unitary amicable pair, each member of which has 317 digits.
Richard McIntosh notes that the largest prime of the form (2^4p + 1)/17 has p = 317, but that (2^4p + 1)/17 is composite for all prime p with 317 < p < 10000. Nobody knows if there are any more such primes. [Post]
The artist Norman Rockwell painted 317 covers for "Saturday Evening Post" magazine. [Dobb]
The smaller of only two 3-digit primes p such that 2^p+p is prime. The other is 701. [Loungrides]
(There are 3 curios for this number that have not yet been approved by an editor.)