Start with the square numbers 1, 4, 9, 16, 25, 36, and so on. Pick any other number and you can express it as a sum of squares. For example, 10 = 1 + 1 + 4 + 4 and 30 = 1 + 4 + 9 + 16. In 1770, French mathematician Joseph-Louis Lagrange proved that every positive integer is either a square itself or the sum of two, three, or four squares. No more than four squares, x2 + y2 + z2 + t2, are ever needed to express any number, no matter how large.
Given Lagrange’s result, number theorists asked whether there are other such expressions, called quadratic forms, that also represent all positive integers. In 1916, Indian mathematician Srinivasa Ramanujan uncovered 53 such expressions. He showed, for instance, that every number could be written as a square plus twice a square plus three times a square plus five times a square.