(show me this)---Every natural number can be written as the sum of four integer squares. --
Lagrange, 1770
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Lagrange's Four Square Theorem states that every natural number can be written as the sum of four integer squares. This fundamental result in number theory was proven by Joseph-Louis Lagrange in 1770. The theorem guarantees that for any positive integer n, we can always find integers a, b, c, and d such that n equals a squared plus b squared plus c squared plus d squared.
Let's examine concrete examples to build intuition. For n equals 1, we have 1 squared plus three zeros. For n equals 2, we need 1 squared plus 1 squared. The number 3 requires three unit squares. Notice that 4 is a perfect square, so it needs only 2 squared. As we continue, we see that 7 requires all four squares: 2 squared plus three 1 squareds. These examples show the theorem works for small numbers and reveal interesting patterns.
Let's analyze numbers by how many squares they require. Perfect squares like 16 need only one square plus three zeros. Some numbers like 13 can be expressed as sums of two squares: 2 squared plus 3 squared. Others like 6 require three squares. Interestingly, certain numbers like 7 require exactly four squares and cannot be expressed with fewer. This analysis shows why Lagrange's bound of four squares is both necessary and sufficient.
Certain numbers require exactly four squares and demonstrate why Lagrange's bound is necessary. These are numbers of the form 4 to the k times 8m plus 7. For example, 7 equals 1 squared plus 1 squared plus 1 squared plus 2 squared. We can verify that 7 cannot be written as a sum of two or three squares through systematic checking. Similarly, 15 and 23 require all four terms. These challenging cases prove that four squares are sometimes essential, not just sufficient.
Lagrange's proof follows a three-step strategy. First, he uses Euler's identity from 1748, which shows that the product of two sums of four squares is also a sum of four squares. This multiplicative property is crucial. Second, he proves the theorem specifically for prime numbers. Finally, he extends the result to all natural numbers using the multiplicative property. The key insight involves quaternion algebra, which provides the mathematical framework for understanding why four squares are always sufficient.