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 12 ways (Posted on 2011-09-08)
Find the lowest number that can be represented as a sum of 2 squares of distinct integers in 12 different ways.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 Solution Comment 4 of 4 |

The answer is 160225, see A025320 in Sloane. Explanation:

I.  See  the solution to The Garden of Pythagoras for a basic explanation about deploying Pythagorean Primes to generate sums of squares.

II.  Let b exceed a.

III. Then:
a^2+b^2=5: 1 way
a^2+b^2=5*13: 2 ways
a^2+b^2=5*13*17: 4 ways
a^2+b^2=5*13*17*29: 8 ways
But these only help if we need exactly (or 'at least') 2^n ways

IV. However, we could also have:
a^2+b^2=5^2
1 way
a^2+b^2=5^3
2 ways
a^2+b^2=5^4
2 ways
a^2+b^2=5^5
3 ways
a^2+b^2=5^6
3 ways
etc.

V.  Alternatively we could have:
a^2+b^2=5^2*13
3 ways
a^2+b^2=5^2*13^2
4 ways
a^2+b^2=5^2*13*17
6 ways
a^2+b^2=5^2*13^2*17
9 ways
a^2+b^2=5^2*13*17*29
12 ways

VI.   5^2*13*17*29= 160225, which must be a minimum solution, since there are no smaller available factors we could substitute for these.

Edited on September 10, 2011, 10:22 am
 Posted by broll on 2011-09-08 20:52:11

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