All entries for Wednesday 16 April 2008

April 16, 2008

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.

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