Lagrange proved in 1770 that every number can be written as the sum of 4 square numbers.

Every number greater than 188 can be written as the sum of at most 5 distinct square numbers.

The sum of the reciprocals of square numbers is  .

The first square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many square numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.