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]
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]