Find the lowest number that can be represented as a sum of 2 squares of distinct integers in 12 different ways.

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