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Topic review (newest first)

noelevans
2013-01-30 08:03:22

Considering just positive integers, there are 6 factorizations of 495 into two factors:
1*495, 3*165, 5*99, 9*55, 11*45 and 15*33.  Each of these corresponds to one of
the six (x,y) pairs bobbym listed in post #4.  For example: 1*495=248^2-247^2 and
24^2-9^2 = (24+9)(24-9) = 33*15.

In general for an odd composite number each of its unique factorizations (other than a perfect
square factorization) corresponds to a difference of squares.  For 9 = 1*9 we get 5^2-4^2
= (5+4)(5-4) = 9*1 but 3*3 has no difference of squares representation unless we allow zero:
3^2-0^2 = (3+0)(3-0) = 3*3.

But we were talking about POSITIVE integers.

If M is an odd composite number and M=n*m where n and m are different, we get

                       

  as a difference of  squares factorization.

If I recall correctly this was involved in one of Fermat's methods of factoring odd composites.

Have a grrreeeeaaaaaaat day! smile

bobbym
2013-01-29 17:09:35

You did not say positive numbers or not. There are 24 positive and negative.

Only 6, just positive.

cooljackiec
2013-01-29 15:12:56

Exactly how many are there, I did 24, but it is wrong..

bobbym
2013-01-29 12:19:25

Hi;

You did not say only positive integers so



are all solutions of x^2 - y^2 = 495

cooljackiec
2013-01-29 11:47:37

A number is called a perfect square if it is the square of an integer. How many pairs of perfect squares differ by 495? (Order does not matter. So, the pair "16 and 9" is the same as "9 and 16".)

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