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

Thanks in advance!

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    $\begingroup$ 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. $\endgroup$
    – amWhy
    Jan 16, 2013 at 2:59
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    $\begingroup$ Ohhhhhh I see, gotcha thanks for the tips yo! $\endgroup$
    – papercuts
    Jan 16, 2013 at 3:01

9 Answers 9

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

13 squares

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

enter image description here

Step 3: fill in the blank space

enter image description here

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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    $\begingroup$ +1 nice straight lines and corners, no wobbliness or smudges. $\endgroup$ Dec 21, 2012 at 13:53
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    $\begingroup$ It's great to see a visual presentation. I never thought about it this way. $\endgroup$
    – krikara
    Dec 21, 2012 at 22:32
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    $\begingroup$ Note $\ $ One can find many analogous visual proofs in Conway and Guy's The Book of Numbers. Highly recommended for budding mathematicians. $\endgroup$
    – Math Gems
    Feb 13, 2013 at 18:42
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    $\begingroup$ @Eric, I seem to recall an answer of yours that also does this... :) $\endgroup$ May 18, 2013 at 18:47
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    $\begingroup$ @orlandpm how did you do the illustration? i love it! $\endgroup$ Aug 26, 2013 at 18:43
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Hint: Consider the difference of two consecutive squares. What is $(k+1)^2-k^2$?

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    $\begingroup$ Oh, haha well that was easy, but how'd you know to use consecutive squares? $\endgroup$
    – papercuts
    Dec 21, 2012 at 7:57
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    $\begingroup$ @papercuts Probably mostly experience. There's not much substitute for it. $\endgroup$ Dec 21, 2012 at 8:00
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    $\begingroup$ It's also one of the simplest sorts of differences of two squares to analyze. $\endgroup$
    – user14972
    Dec 21, 2012 at 9:35
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    $\begingroup$ @papercuts Try it with the difference of any two squares: $(k+n)^2-k^2$. Then let n=1. $\endgroup$
    – Griffin
    Dec 21, 2012 at 18:10
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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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    $\begingroup$ Would the downvoter care to explain? $\endgroup$
    – P.K.
    May 18, 2013 at 14:32
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    $\begingroup$ nicely proved ! $\endgroup$
    – Debashish
    Jun 20, 2014 at 8:34
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Here is a more calculational answer where we try to 'construct' the solution by minimizing the amount of 'magic' in the proof. I'm assuming all variables are integers.

Given an odd $\;n\;$, we are asked to find $\;a,b\;$ such that $$ a^2 - b^2 = n $$

Let's calculate: \begin{align} & a^2 - b^2 = n \\ \equiv & \;\;\;\;\;\text{"arithmetic -- the simplest thing I know about the difference of squares"} \\ & (a+b) \times (a-b) = n \\ \equiv & \;\;\;\;\;\text{"arithmetic -- the simplest thing to give both sides similar structure"} \\ & (a+b) \times (a-b) = n \times 1 \\ \Leftarrow & \;\;\;\;\;\text{"logic: Leibniz -- the simplest thing to exploit the structural similarity"} \\ & a+b = n \;\land\; a-b = 1 \\ \equiv & \;\;\;\;\;\text{"arithmetic: solve for $\;a,b\;$"} \\ & a = (n+1)/2 \;\land\; b = (n-1)/2 \\ \end{align} Now since $\;n\;$ is odd, both $\;(n+1)/2\;$ and $\;(n-1)/2\;$ are integers, and therefore we have found the required $\;a,b\;$.

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    $\begingroup$ Could we please explain the Leibnitz step? I'm not sure how a + b = n and a - b = 1 follows from (a+b)×(a−b)=n×1. $\endgroup$
    – mcandre
    Jan 2, 2016 at 2:12
  • $\begingroup$ There are plenty of operators (e.g. modulo) where f(a, b) = f(c, d) has solutions where (c, d) != (a, b). Is multiplication special this way? Would a + b = 1 and a - b = n also have integer solutions, since n*1 = 1*n? $\endgroup$
    – mcandre
    Jan 2, 2016 at 2:28
  • $\begingroup$ You seem to be confusing the arrow directions. This step says that if $\;a+b=n\;$ and $\;a-b=1\;$, then also the product of the left hand sides equals the product of the right hand sides. Leibniz' rule just says in general that $\;x = y\;$ implies $\;\ldots x \ldots \;=\; \ldots y \ldots\;$. The philosophical name is "indiscernibility of identicals". $\endgroup$ Jan 2, 2016 at 8:57
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The algebraic equivalent of orlandpm's very elegant illustration is this:

Let a be any integer . . .

$K = (a+1)^2 - a^2$

$K = (a^2 + 2a + 1) - a^2$

$K = a^2 + 2a + 1 - a^2$

$K = 2a + 1$, which defines any odd integer; Q.E.D.

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    $\begingroup$ Eric had already hinted that. And bringing old posts up by answering them without bringing something really new isn't something you should do. Also, this site uses latex math-mode to format maths. (Just put $s around your formulas) $\endgroup$
    – xavierm02
    Jul 27, 2013 at 23:59
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To get the complete classification of differences of Thai squares, we use the elementary observation that $a^2-b^2=(a+b)(a-b)$.

Clearly, $a+b$ and $a-b$ at either both even or both odd. So a d difference of two squares is either of our a multiple of $4$.

In fact, if $N$ is odd or a multiple of $4$, then we can always find $a$, $b$ such that $N=a^2-b^2$: if $N$ is odd, take $a=(N+1)/2$ and $b=(N-1)/2$, and if $N$is sa multiple of $4$, take $a=N/2+1$ and $b=N/2-1$.

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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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    $\begingroup$ Re "Since B and k are both just constants, they can be treated as equal": no: you already fixed both. $\endgroup$
    – msh210
    Dec 21, 2012 at 15:30
  • $\begingroup$ @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). $\endgroup$ Dec 21, 2012 at 15:37
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You could add.

For mod ((2x+1),4)= 1, 2x+1 has at least one more solution than (k^2+1)-k^2=2x+1 and k^2 will be odd and (k-n)^2 will be even. Opposite if Mod((2x+1),4)= 3. n is always odd.

k^2-(k+n)^2= 2x+1, if 2x+1 is a composite number, if not, 2x+1 only has the solution (k^2+1)-k^2=2x+1 and 2x+1 is a prime number.

Sorry Mathjax is not working on my computer.

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    $\begingroup$ Who are $x$ and $n$? The problem speaks only about $k$. What do you mean by "solution", given that there are no equations involved either in the question, or in your own answer? $\endgroup$
    – Alex M.
    Nov 20, 2015 at 10:52
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Proof for odd numbers: Divide any odd number, $x$, in half, then let $C$ be the half rounded up and $B$ be the half rounded down. Then, $C^2 - B^2 = x$. Let's examine this algebraically:

$$C = \frac{x+1}{2}$$ $$B = \frac{x-1}{2}$$ $$C^2 - B^2 = \biggl(\frac{x+1}{2}\biggr)^2 - \biggl(\frac{x-1}{2}\biggr)^2 = \frac{x^2+2x+1}{4} - \frac{x^2-2x+1}{4} = \frac{x^2-x^2+2x+2x+1-1}{4} = \frac{4x}{4} = x$$

Then, because even/odd integers alternate, $x \pm 1$ will always result in an even integer when $x$ is odd and an odd integer when $x$ is even.

Because only even integers are divisible by two (without incurring a decimal), $x \pm 1$ must be even to be divisible by 2, so $x$ must be odd. Otherwise, if $x$ were even, then $C$ and $B$ would not be integers.

Example for odd numbers: Take the number 31. Let $C=\frac{31+1}{2}=16$ and $B=\frac{31-1}{2}=15$. Then, $16^2 - 15^2 = 256 - 225 = 31$.

Proof for all composite numbers: every pair of $x$'s factors, $p$ and $q$, including composite factor pairs, where the oddity of $p$ matches the oddity of $q$, corresponds to a separate representation as the difference of two squares:

$$p \times q = x$$

$$C = \frac{p + q}{2}$$

$$B = \frac{|p - q|}{2}$$

$$C^2 - B^2 = \biggl( \frac{p + q}{2} \biggl)^2 + \biggl( \frac{|p - q|}{2} \biggl)^2 = \frac{p^2 + 2pq + q^2}{4} - \frac{p^2 - 2pq + q^2}{4} = \frac{p^2 - p^2 + q^2 - q^2 + 4pq}{4} = pq = x$$

Example for every-other even numbers: Because half of every-other even number (that is, 4, 8, 12, etc.) has an even oddity, it can be shown that at least every-other even number can be expressed as the difference of two squares. For $x=48$, we get $p=2$ and $q=\frac{x}{p}=24$. Then, $C=\frac{2+24}{2}=13$ and $B=13-2=11$. We verify $C^2 - B^2 = 169 - 121 = 48$.

Conclusion: The only numbers that cannot be expressed as the difference of two squares are the sequence $4n+2$: 2, 6, 10, 14, 18, 22, etc.

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