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=196-9 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 = k²-j² then by factoring, ab = (k+j)(k-j).

If we let k+j=a and k-j=b, adding the equations we get

2k=a+b

subtracting the equations we get

2j=a-b

so

k=(a+b)/2 and j=(a-b)/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=(5-3)/2, that is, 4 and 1, so 4²-1²=16-1=15, as needed.

Posted by Charlie on 2003-04-16 08:29:39