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.
Starting with the example given, note that (11+17)/2 = 14, and (17-11)/2 = 3, which numbers correspond to the square roots of 196 and 9.
Start with odd numbers, (2x-1) and (2y-1) whose product is (2x-1)(2y-1). Multiplying out: (4xy-2x-2y+1) (I)
Using the earlier insight, our 'guess squares' are ((2x-1)+(2y-1))/2, or (x+y-1) and ((2x-1)-(2y-1))/2, or (x-y)
(x+y-1)^2 = x^2+2xy-2x+y^2-2y+1 and (x-y)^2 = x^2-2xy+y^2.
x^2+2xy-2x+y^2-2y+1-(x^2-2xy+y^2) = (4xy-2x-2y+1) (I) as surmised.
For even numbers 2x and 2y, 2x*2y = 4xy (II).
Our new 'guess squares' are (2x+2y)/2, or (x+y) and (2x-2y)/2, or (x-y). (x+y)^2-(x-y)^2 = 4xy (II), again confirming the relation.
The two perfect squares, for either parity, are then a = ((x1+y1)/2)^2, b= ((x1-y1)/2)^2, where x1 and y1 are positive integers having the same parity.
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Posted by broll
on 2023-07-13 23:43:58 |
Possible Solution
