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.
(In reply to re: solution
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 = p²-q² then by factoring, kj = (p+q)(p-q).
If we let p+q=k and p-q=j, adding the equations we get
subtracting the equations we get
p=(k+j)/2 and q=(k-j)/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=(5-3)/2, that is, 4 and 1, so that a = p² = 16 and b = q² = 14²-1²=16-1=15, as needed.
Posted by Charlie
on 2003-04-16 10:18:56