Arithmetic Sequence of Four Squares

Investigation

For me, this all started from an innocent-looking question on an Internet message board: is it possible to find four perfect squares in arithmetic sequence?  Right away, I found a few arithmetic sequences of three squares, such as 1, 25, 49.  Soon (and with some help, I'll admit), I was able to generate an arbitrary number of such sequences (the Preamble, below, explains how).  I posted all this stuff on this website, and, thanks to Google, people found it, and gave me lots of other tidbits of information -- tidbits such as the fact that this question (along with the result that there are no arithmetic sequences of four squares) is attributed to Fermat.

I found thousands of arithmetic sequences of three squares, but none of four squares.  I noticed that the constant differences are multiples of 24 in all the arithmetic sequences of squares, so I tried to use that factoid to prove that the fourth number in the sequence wasn't a square.  I looked for parity arguments to no avail.  I finally gave up, and searched the Internet, and came up with the first proof.  Then a reader of my website showed me the second proof.  If you have any thoughts on this topic, please send me an email by clicking on the email link at the bottom of this page.

Preamble, Lemmas, and Stuff

Before I get to the proofs themselves, there are some things you should know.  Basic Number Theory stuff, such as the GCD are introduced in this section.  Then I show you how to generate Primitive Pythagorean Triples quickly and easily, from any pair of coprime opposite-parity positive integers.  (And I define all those pesky terms I just used, too.)  Finally, I lay out a fascinating relationship between Pythagorean Triples and arithmetic sequences of three squares.  Let's get started!

Proof 1

Here is Proof 1.

Proof 2

Here is Proof 2.

Internet References

none.

Related Pages in this website

Arithmetic Sequence of Squares -- a series of rambling discussions about the possible existence of an arithmetic sequence of four squares

The webmaster and author of the Math Help site is Graeme McRae.
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