Lagrange's Four-square Theorem - Generalizations

Generalizations

Lagrange's four-square theorem is a special case of the Fermat polygonal number theorem and Waring's problem. Another possible generalisation is the following problem: Given natural numbers a, b, c, d, can we solve

n = ax₁2 + bx₂2 + cx₃2 + dx₄2

for all positive integers n in integers x₁, x₂, x₃, x₄? The case a = b = c = d = 1 is answered in the positive by Lagrange's four-square theorem. The general solution was given by Ramanujan. He proved that if we assume, without loss of generality, that a ≤ b ≤ c ≤ d then there are exactly 54 possible choices for a, b, c, d such that the problem is solvable in integers x₁, x₂, x₃, x₄ for all n. (Ramanujan listed a 55th possibility a = 1, b = 2, c = 5, d = 5, but in this case the problem is not solvable if n = 15.)