Math Is Fun Forum

cooljackiec

Member

Registered: 2012-12-13

Posts: 186

squares

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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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: squares

Hi;

You did not say only positive integers so

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

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

cooljackiec

Member

Registered: 2012-12-13

Posts: 186

Re: squares

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

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I see you have graph paper.
You must be plotting something

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: squares

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

Only 6, just positive.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

noelevans

Member

Registered: 2012-07-20

Posts: 236

Re: squares

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!

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Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.