[seqfan] Re: puzzle

Chris Thompson cet1 at cam.ac.uk
Sat Feb 6 21:31:54 CET 2016


On Feb 3 2016, Harvey P. Dale wrote:

>>The sum of N consecutive square numbers is a square.
>>What can N be?

>If the sequence has to start with 1, then 1 and 24 are terms, but there
>are no further terms up to 1 million.

I shouldn't bother trying any further... :-)

This is Lucas' "Square Pyramid Puzzle" which is really quite famous. See
for example section D3 in Richard Guy's "Unsolved Problems in Number Theory".
You can find a (technically) elementary proof that there are no other
solutions in

  Anglin, W. S. (1990). The Square Pyramid Puzzle. The American 
  Mathematical Monthly, 97(2), 120-124. http://doi.org/10.2307/2323911 

For more amusement, you can find the fact that (0,1,2,...,23,24|70) is
a null vector in 25+1 dimensional Lorentzian space used to interesting
effect in chapters 26-28 of Conway & Sloane's "Sphere Packings, Lattices
and Groups".

-- 
Chris Thompson
Email: cet1 at cam.ac.uk




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