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.
(In reply to
re: solution by jennifer)
Oh, Oh, I used misleading letters, not using a and b the way the problem called for. I write below what I should have had:
Let the required a = p² and b = q² where p and q are integers so that a and b would be perfect squares.
If kj = p²q² then by factoring, kj = (p+q)(pq).
If we let p+q=k and pq=j, adding the equations we get
2p=k+j
subtracting the equations we get
2q=kj
so
p=(k+j)/2 and q=(kj)/2
If k and j are both odd or both even, the numerators will be even, so p and q will be integers as required.
So, for example, if we seek 3*5=15, we can use p=(5+3)/2 and q=(53)/2, that is, 4 and 1, so that a = p² = 16 and b = q² = 14²1²=161=15, as needed.

Posted by Charlie
on 20030416 10:18:56 