4159

This number is a prime.

4159² - 1 is one of Ramanujan's highly composite numbers. Note that the famous Hardy-Ramanujan number (1729) is the first four digits of 4159². [Gupta]

The arithmetic mean of the two successive Mersenne primes 2^7-1 and 2^13-1. It is the largest known prime with this property. [Earls]

The first 4-digit prime to occur in the decimal expansion of π. [Loungrides]