# 317

This number is a prime.

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} + 1^{3} + 7^{3}. [Dobb]

317# - 1 is a primorial prime. [Dobb]

A door with room number 317 appears in film footage taken of Lee Harvey Oswald shortly before his death.

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.

Former pitching ace Walter "Big Train" Johnson once hurled a coin 317 feet across the Rappahannock River in Virginia.

According to the Australian Bureau of Meteorology's *Annual Climate Summary*, 317 millimetres is the average annual rainfall over Australia for Australia's driest year since 1890 (1902).

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]

The upperbound for Skewes's number has been improved by Patrick Demichel to 317 digits.

There are currently 317 recognized species of turtles and tortoises in the world. [Buhlmann]

After 1 and 2, n=317 is the smallest case in which n divides the prime index (6974) of the nth Honaker prime (70379). [Gaydos]

It is not known whether π(x) is ever greater than li(x) for any number less than 317 digits, assuming the Riemann hypothesis. [Zegowitz]

317 of the first thousand prime numbers have no digits in common with their prime indices. [Gaydos]

The smallest prime which is the sum of three consecutive sphenic numbers, i.e., 102 + 105 + 110. Note that 102 = 2 × [3*17]. [Worrom]