Any product of two evens or two odds (sticking just to positives for the purpose of this problem) can be expressed as a difference of two perfect squares. 11*17=187=1969 is an example.
A: Prove this idea.
B: Come up with a formula that gives the two perfect squares. Call the larger one a and the smaller one b.
If ab = k²j² then by factoring, ab = (k+j)(kj).
If we let k+j=a and kj=b, adding the equations we get
2k=a+b
subtracting the equations we get
2j=ab
so
k=(a+b)/2 and j=(ab)/2
If a and b are both odd or both even, the numerators will be even, so k and j will be integers as required.
So, for example, if we seek 3*5=15, we can use k=(5+3)/2 and j=(53)/2, that is, 4 and 1, so 4²1²=161=15, as needed.

Posted by Charlie
on 20030416 08:29:39 