1031
This number is a prime.
The number of ones that form the fifth and largest known repunit prime. It was first proven prime in 1985. (The next few that could possibly be prime are those with 49081, 86453, 109297, and 270343 digits.)
1031 = 2*3*5 + 7*11*13. [Capelle]
Halloween falls on "ten thirty-one" (10/31) each year. It is a cross-quarter date, approximately midway between an equinox and a solstice.
The final bid for an Erdős number on eBay (April 30, 2004) was $1031. Note that the bidding was for an Erdős number of 5. [McCranie]
The lesser of the smallest twin emirps. Their reversals are also twins and reflective. [Punches]
The only emirp among the subscripts
p of all known repunit primes
Rp = (10p -1)/9, as of August 2009. [Beedassy]
Simon Plouffe collected and published 1031 formulas that were generated by a computer program in 1992.
Starts the first group of four emirps formed from the same
digits: 1031, 1301, 1103, 3011. [Silva]
The smallest 4-digit emirp p whose the reversal of square
is equal to the square of the reversal of p, i.e.,
1031^2=1062961 and 1301^2=1692601. [Loungrides]
The number of cells of the male C. elegans -- a
nematode worm, small enough that you can barely see an
adult unless it’s placed right up against your eye. [Post]
One approach to "quantum gravity" is string theory: a mathematical description of particles and forces at scales
1031 times smaller than a proton. [Annemberg]