# Prove every odd integer is the difference of two squares

I know that I should use the definition of an odd integer ($2k+1$), but that's about it.

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 Hi, papercuts. You might want to start thinking about upvoting answers that are helpful (click on the upward arrow (grey) to the left of the answer. You can also accept one answer, which you can do by clicking on the "greyed-out" check-mark to the left of the answer you want to accept. upvoting and accepting answers encourages people to take the time needed to answer your questions, and is a way to show appreciation. – amWhy Jan 16 at 2:59 Ohhhhhh I see, gotcha thanks for the tips yo! – papercuts Jan 16 at 3:01

Step 1: pick an odd number (like $n=13$ here)

Step 2: bend it in "half" (any odd number $n$ can be written as $2k+1$, and $13=2\cdot 6 + 1$)

Step 3: fill in the blank space

Step 4: Count squares. (Here, the blue square has area $36=6^2$, while the whole square has area $49=7^2$)

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+1, very nice illustration. – Eric Dec 21 '12 at 8:27
+1 Very very nice! – Nameless Dec 21 '12 at 8:33
+ Very nice colour – Grijesh Chauhan Dec 21 '12 at 12:09
+1 nice straight lines and corners, no wobbliness or smudges. – Graham Borland Dec 21 '12 at 13:53
It's great to see a visual presentation. I never thought about it this way. – krikara Dec 21 '12 at 22:32

Hint: Consider the difference of two consecutive squares. What is $(k+1)^2-k^2$?

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Oh, haha well that was easy, but how'd you know to use consecutive squares? – papercuts Dec 21 '12 at 7:57
@papercuts Probably mostly experience. There's not much substitute for it. – Alex Becker Dec 21 '12 at 8:00
It's also one of the simplest sorts of differences of two squares to analyze. – Hurkyl Dec 21 '12 at 9:35
@papercuts Try it with the difference of any two squares: $(k+n)^2-k^2$. Then let n=1. – Griffin Dec 21 '12 at 18:10

HINT: \begin{align} &2k + 1 \\= & 1\cdot(2k + 1) \\ =& \left(k + 1 - k \right)\left(k + 1 + k\right) \\ = & \cdots\end{align}

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 Would the downvoter care to explain? – Πάρτη Κοηλί 2 days ago

Eric and orlandpm already showed how this works for consecutive squares, so this is just to show how you can arrive at that conclusion just using the equations.

So let the difference of two squares be $A^2-B^2$ and odd numbers be, as you mentioned, $2k+1$. This gives you $A^2-B^2=2k+1$.

Now you can add $B^2$ to both sides to get $A^2=B^2+2k+1$. Since $B$ and $k$ are both just constants, they could be equal, so assume $B=k$ to get $A^2=k^2+2k+1$. The second half of this equation is just $(k+1)^2$, so $A^2=(k+1)^2$, giving $A = ±(k+1)$, so for any odd number $2k+1$, $(k+1)^2-k^2=2k+1$.

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Re "Since B and k are both just constants, they can be treated as equal": no: you already fixed both. – msh210 Dec 21 '12 at 15:30
@msh210 Neither of them are "fixed" yet; if they were, they would have set values and wouldn't be very usable for proofs (and, barring certain specific values, would probably be referred to with that value instead). I edited my answer to try and clarify (since it's more of an assumption than a known). – MyNameIsNotMcThomasJohannson Dec 21 '12 at 15:37