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For example,

3 = 2^2 - 1^2
5 = 3^2 - 2^2
7 = 4^2 - 3^2
 ...

Is there a general formula to explain this phenomenon?

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  • $\begingroup$ $(x+1)^2-x^2=x^2+2x+1-x^2=2x+1$. $\endgroup$ Sep 10, 2014 at 7:17
  • $\begingroup$ Its generalize .. not generiloze :) $\endgroup$
    – Sawarnik
    Dec 9, 2014 at 9:10

3 Answers 3

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The difference of two general squares doesn't need to be an odd number: $16$ and $4$ are both square numbers, but there difference $16-4=12$ is not odd.

Is there a general formula to explain this phenomenon?

Yes, if the question is, why is the difference between two consecutive square numbers odd? Two consecutive square numbers are of the form $n^2, (n+1)^2$ and their difference is $(n+1)^2 - n^2 = n^2 + 2n+ 1 - n^2 = 2n+1$, and $2n+1$ is an odd number.

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If you have a number $n\geq0$ then: $$(n+1)^2 - n^2 = (n^2+2n+1)-n^2 = 2n+1,$$ which makes all odd numbers (i.e. every odd number can be written as $2n+1$, with a certain value for $n$).

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The difference of an odd number and an even number (or the other way around) is an odd number.

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