### Number discoveries

Things I didn’t know about numbers until very recently:-

- Primes of the form
*4n + 3*are never the sum of two squares. - Primes of the form
*4n + 1*have one and only one way of being the sum of two squares. - 8 and 144 (and 1, but that doesn’t really count) are the only cube and square (or any other exact power) in the Fibonacci sequence. But while it is easy to demonstrate this empirically for very large numbers of values in the sequence, the proof has turned out to be surprisingly tricky. In fact, it was only after Andrew Wiles proved Fermat’s Last Theorem that it became possible, ten years later, to exploit some of his techniques to prove that there are no perfect-power numbers in the
*F*sequence other than 8 and 144.

## 4 comments by 3 or more people

## Steve Rumsby

If you enjoy such facts, you’ll enjoy reading The Dictionary of Curious and Interesting Numbers

17 Apr 2008, 07:18

## Trevor Hawkes

In case you weren’t aware, the theorem that 8 and 144 are the only pure powers in the Fibonacci sequence was proved by our Warwick colleague Samir Siksek.

20 Apr 2008, 13:46

## John Dale

I didn’t know that; awesome! It’s one of those theorems, a bit like Fermat’s Last, I suppose, which sounds so simple when it’s stated as a problem that it’s bewildering to hear that the proof turns out to be so complex.

21 Apr 2008, 11:28

1. Gödel’s incompleteness theorem says that within a given logical system there are true results that cannot be proved. So perhaps the most amazing thing is that proofs exist at all for FLT and the like. And mathematicians never know for sure whether there’s a one-line proof and they’ve missed it.

2. BTW, there’ s a “one-sentence” proof (well … sort of) of the sum of 2 squares result near the bottom of this page, due to Zagier out of Heath-Brown. The Gaussian integers also provide a good insight into the result (see the two proofs by Dedekind higher up the page).

21 Apr 2008, 13:15

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