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MATH 035
Penn State University
Dr. James Sellers
Handout: Sums of Two Squares
Consider the following: We know that there are lots of triples of numbers (a,b,c) such than
a 2 + b 2 = c 2 thanks to the Pythagorean Theorem. What if we do something a bit different?
Instead of forcing the right-hand side to be a square (like c 2 ), we simply put any integer there
and ask whether that integer can be written as a sum of two squares? Said a different way, if
we choose some integer N, can we find numbers a and b such that a 2 + b 2 = ? It turns out
that there is a very nice solution to this problem which we will investigate today.
So let’s start with replacing N by a prime number and see what happens. Clearly, N = 2 has a
solution if we just let a = 1 and b = 1. What about the odd primes? Well, N = 3 cannot be
written as a sum of two squares, but N = 5 can be written as a sum of two squares, namely 4 + 1
(let a = 2 and b = 1). Now let’s spend some time determining which primes have solutions and
which ones don’t.
3 _____________
5 _____________
7 _____________
11 _____________
13 ____________
17 ____________
19 ____________
23 _____________
29 ____________
31 ____________
37 ____________
41 _____________
43 ____________
47 ____________
53 ____________
59 _____________
61 ____________
67 ____________
71 ____________
73 _____________
79 ____________
83 ____________
89 ____________
97 _____________
So which primes appear to be representable as the sum of two squares?
Albert Girard was the first to make the observation (in 1632) and Fermat was first to claim a proof of it.
Fermat announced this theorem in a letter to Marin Mersenne dated December 25, 1640.
OK, that’s cool. But what about the other integers? Can something be said about when a nonprime number can be represented by a sum of two squares? Well, let’s again work on several
examples to see if a pattern emerges. Here I have provided a smattering of non-prime
numbers. In this case, do the following:
(a) Compute the prime factorization of each number, and
(b) Determine which numbers can be written as a sum of two squares.
15 _______________ _______________
45 _______________ _______________
63 _______________ _______________
65 _______________ _______________
80 _______________ _______________
96 _______________ _______________
102 ______________ _______________
104 ______________ _______________
106 ______________ _______________
117 ______________ _______________
136 ______________ _______________
180 ______________ _______________
185 ______________ _______________
187 ______________ _______________
200 ______________ _______________
245 ______________ _______________
So what pattern seems to emerge when determining whether a non-prime number can be
written as a sum of two squares?
The first proof of this result appears to have been found by Euler and is based on infinite
descent. He announced it in a letter to Goldbach on April 12, 1749.
Two other facts of importance are worth noting before we close:
1. You should note that some integers can be expressed as a sum of two squares in more than
one way; for example, 50 = 49 + 1 and 50 = 25 + 25. Also, 65 = 64 +1 and 65 = 49 + 16.
2. What about those numbers that can’t be expressed as a sum of two squares? Well, what if
we give them more squares to work with? It turns out that EVERY number can be expressed as
a sum of FOUR squares. The first proof of this result is due to Lagrange in 1770.
© 2010, James A. Sellers