Sums of squares
By Alan Crowe (Fri Apr 17, 2009 at 03:10:33 PM EST) fermat (all tags)

Some numbers can be written as the sum of two squares. For example 113=7²+8² and 205=14²+3².

If we multiply 113×205=23165 is the result a sum of squares? Yes, 23165=146²+43², but why? 146 and 43 seem unrelated to any of 3,7,8,14.

I think I have a proof that the product of sums of two squares is always a sum of two squares. It is two short and cunning lines of algebra. You may wish to treat this as a puzzle and try proving it yourself. However, larger numbers are often sums of two squares in many ways. My algebra expresses 23165 as 91²+122².

If you were a seventeenth century French jurist it might occur to you to ask about numbers that were the sum of a square and twice a square: x²+2y². For example 113=9²+2×4² and 205 is bugger it. 205 doesn't have the form x²+2y². Well that is kind of the point. It is because only some numbers have this form that it is interesting to ask if multiplication preserves it. 219=13²+2×5². What about 113×219=24747? It is 157²+2×7².

I have two more cunning lines of algebra to prove the point. Puzzle number two!

What about the sum of a square and three times a square? For example 97=7²+3×4² and 111=6²+3×5². In conformity with the emerging pattern 97×111=10767=102²+3×11²

Puzzle number three is infinitely harder. For each n is it true that all numbers of the form x²+ny² preserve this form under multiplication? This is infinitely harder because the proof is for all n, not just 1,2 and 3. If you have already solved puzzles one and two you may find the infinitely harder puzzle a little easier than the first two.

The story behind this diary entry is that last saturday a little red sticker caught my eye. Half price! Who could resist? Harold M. Edwards' celebrated book Fermat's Last Theorem: A genetic Introduction to Algebraic Number Theory for only £16. Complete with the inside story on the events of 1847.

Now I must dally over the three easy chapters that open the book. My experience with mathematics books is that they rapidly become too hard for me, but sufficient unto the day is the mathematics thereof.

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Attempt at #3 by stevew (2.00 / 0) #1 Sat Apr 18, 2009 at 04:10:11 AM EST
Posted on behalf of my 11 year old who you kept amused for a little while this afternoon.
And he added the comment 'for very low values of infinity'.

Do your own homework! by Alan Crowe (2.00 / 0) #2 Sat Apr 18, 2009 at 01:53:03 PM EST

Getting your mini-Tao to do it for you is cheating :-)

I'm working on another diary entry that explains where the idea for sticking in 2abcd-2abcd comes from. Then you won't have to ask and be told "Aw Dad, it's just obvious!"

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