Four Square Theorem

September 7, 2011

It seems that every educated person has heard of Fermat's Last Theorem, although it's likely that only the mathematically-minded can state what it is. As I wrote in a previous article (You Can Get There From Here, March 26, 2008), Fermat's Last Theorem simply states that for integers n>2, the equation,

aⁿ + bⁿ = cⁿ

has no solutions other than a=b=c=0. Princeton University mathematicians Andrew Wiles and Richard Taylor proved the theorem was true in 1994, more than 350 years after it was stated. The conjecture was stated by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica by Diophantus. Fermat claimed he had a proof that was too large to fit in the margin.

There's a distant cousin to this theorem that's found also in Arithmetica. It's not quite as difficult. Lagrange proved it in 1770, and an undergraduate mathematics major can understand the short proof, which can be found here. It's called Lagrange's Four-Square Theorem, and it's simply the fact that every positive integer n can be written as the sum of four squares; that is,

n = a² + b² + c² + d²,

where a, b, c and d are integers. Sometimes the qualification, "at most, four squares," is used. If we allow the square of zero, that qualification isn't needed.

Joseph Louis Lagrange Lagrange is known for much more than his mathematics. He's well known to physicists for the Lagrangian; and to astronomers for the Lagrangian points, to name just two. (Via Wikimedia Commons).

Surprisingly, Euler was not able to prove the theorem himself, although he contributed a piece called the four-square identity.

As usual, the theorem started as a conjecture. The conjecture, called Bachet's conjecture, was stated by Claude Gaspard Bachet de Méziriac, a French mathematician and classics scholar, who translated Diophantus' Arithmetica, with commentary, from Greek to Latin in 1621. A portion of the section of the book that describes the conjecture appears below.

Diophantus Arithmetica, Latin translation by Bachet, 1621, top of page IIII-241. (Via Google Books).[2]

The last line in the figure reads,

Interim libet id inductione confirmare, ostendendo propriu esse numerorum omnium ab 1 usque ad 120, ut constant ex sequenti diagrammate.

In the mean time I want to confirm this by induction, specifically by showing it for all of the numbers from 1 to 120, as is evident from the following table. (My translation)

Bachet then presents a table of examples for all numbers from 1 - 120 that includes the following:

28 = 1² + 1² + 1² + 5²
28 = 1² + 3² + 3² + 3²
28 = 2² + 2² + 2² + 4²
48 = 4² + 4² + 4²
48 = 2² + 2² + 2² + 6²
60 = 1² + 1² + 3² + 7²
60 = 1² + 3² + 5² + 5²
60 = 2² + 2² + 4² + 6²
112 = 2² + 2² + 2² + 10²
112 = 2² + 6² + 6² + 6²
112 = 4² + 4² + 4² + 8²
120 = 2² + 4² + 10²

In 1798, Adrien-Marie Legendre advanced the theorem by a partial proof that only three squares are needed to represent a positive integer, if, and only if, the integer is not of the form 2^2k(8m + 7), where k and m are positive integers. Gauss plugged a gap in Legendre's proof.

One very interesting discovery was made by Jacobi, who found that the number of ways a positive integer n can be represented as a sum of four squares is eight times the sum of the divisors of n, for odd n, and 24 times the sum of the odd divisors, for even n. Note that in order to get Jacobi's count, the numbers being squared can be negative, and you need to count all combinations of numbers; that is,

1 = 1² + 0² + 0² + 0²
1 = 0² + 1² + 0² + 0²
1 = (-1)² + 0² + 0² + 0²
etc...

References:

1. Eric W. Weisstein, "Lagrange's Four-Square Theorem," MathWorld--A Wolfram Web Resource.
2. Claude Gaspar Bachet, translator: "Diophantus of Alexandria, Arithmetica," 1621, via Google Books.