What is the Expression for the Difference of Two Square Numbers?

[chhitiz]

is there an expression for the difference of two square no.s, except, of course for rsquare minus ssquare

 

[huba]

(r+s)(r-s)

 

[chhitiz]

god you are funny. i posted this question because i was trying to find out all possible no.s which have a self-conjugating ferrer's graph, and needed to see if diff. of two square no.s is in anyway in form of a triangle no. or not. anyways, i have inferred that all no.s except for 2 can be jotted as a self-conjugating ferrer's graph. am i correct?

 

[CRGreathouse]

Your first post was entirely unclear, and I'm not quite sure what your second means. "if diff. of two square no.s is in anyway in form of a triangle no. or not"? Does that mean something like "Is the difference of two squares triangular?"?

If that interpretation is generally correct, I can think of at least four ways to take it:
$$\forall n>m \exists t:t(t+1)/2=n^2-m^2$$
$$\forall n \exists m,t:t(t+1)/2=n^2-m^2$$
$$\forall N \exists n>m>N,t:t(t+1)/2=n^2-m^2$$
$$\exists n,m,t:t(t+1)/2=n^2-m^2$$

If it's not, then you'll have to be more clear.

 

[chhitiz]

that was exactly what i meant, so thanks

 

[CRGreathouse]

Which?

 

[chhitiz]

the first one. by the way, am i not correct in saying that all +ve integers except 2 can be expressed as a self conjugating ferrer's graph?

 

[CRGreathouse]

the first one.

$$\forall n>m \exists t:t(t+1)/2=n^2-m^2$$

The first few (n, m) for which this fails are:
(3, 1)
(5, 1)
(3, 2)
(4, 2)
(4, 3)
(5, 3)
(5, 4)