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Division is somewhat more complex than I thought

Let $$a, b, c$$ be all positive integers such that $$a^2 +b^2= c^2$$.

Prove that, for all positive integers $$n$$, both $$a^{2n+1}+b^{2n+1}+c^{2n+1}$$ and $$(b+c)(c+a)(a+b)$$ are divisible by $$a+b+c$$.

Note by Biswajit Barik
7 months, 2 weeks ago

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First let's show that $$ab$$ is divisible by $$a+b+c$$:

$ab = \dfrac{(a+b)^2-a^2-b^2}{2} = \dfrac{(a+b)^2 - c^2}{2} = \dfrac{(a+b+c)(a+b-c)}{2}$

Now $$a+b+c$$ and $$a+b-c$$ differ by an even number, so they have the same parity. However, they can't both be odd, because then the final expression above would not be an integer, even though it's equal to $$ab$$ which is an integer. Therefore $$a+b-c$$ is even, so $$\dfrac{a+b-c}{2}$$ is an integer. Therefore $$ab$$ is divisible by $$a+b+c$$. $$\square$$

For the rest of the proof, for brevity let $$p = a+b+c$$ and let $$q = \dfrac{ab}{a+b+c}$$ (which is an integer by the above lemma).

\begin{align} (b+c)(c+a)(a+b) & = (p-a)(p-b)(p-c) \\ & = p^3 - (a+b+c)p^2 + (ab+ac+bc)p - abc \\ & = (ab+ac+bc)p - pqc \\ (a+b)(b+c)(c+a) & = p(ab+ac+bc - qc) \end{align}

Therefore $$(b+c)(c+a)(a+b)$$ is divisible by $$p$$. $$\square$$

Next we'll show that $$a^{2n+1} + b^{2n+1} + c^{2n+1}$$ is divisible by $$p$$ for any $$n \ge 0$$:

$a^{2n+1} + b^{2n+1} + c^{2n+1} = a^{2n+1} + b^{2n+1} - (a+b)^{2n+1} + \left[(a+b)^{2n+1} + c^{2n+1} \right]$

I say that $$(a+b)^{2n+1} + c^{2n+1}$$ is divisible by $$p$$. Why? Using modular arithmetic, we have $$a+b \equiv -c \pmod{p}$$. Therefore, $(a+b)^{2n+1} + c^{2n+1} \equiv (-c)^{2n+1} + c^{2n+1} \equiv -c^{2n+1} + c^{2n+1} \equiv 0 \pmod{p}$

Hence $$(a+b)^{2n+1} + c^{2n+1} = pr$$ for some integer $$r$$. Continuing with the proof:

\begin{align} a^{2n+1} + b^{2n+1} + c^{2n+1} & = a^{2n+1} + b^{2n+1} - (a+b)^{2n+1} + \left[(a+b)^{2n+1} + c^{2n+1} \right] \\ & = a^{2n+1} + b^{2n+1} - (a+b)^{2n+1} + pr \\ & = pr + a^{2n+1} + b^{2n+1} - \sum_{k=0}^{2n+1} \binom{2n+1}{k} a^{2n+1-k} b^k \\ & = pr - \sum_{k=1}^{2n} \binom{2n+1}{k} a^{2n+1-k} b^k \end{align}

Now let's consider that last sum. For each $$k$$ between $$1$$ and $$2n$$, the exponents of $$a$$ and $$b$$ are always at least $$1$$. Therefore for each $$k$$, $$a^{2n+1-k} b^k$$ is divisible by $$ab$$. Hence the entire sum is divisible by $$ab$$. Let's say the sum equals $$abs$$ for some integer $$s$$.

\begin{align} a^{2n+1} + b^{2n+1} + c^{2n+1} & = pr - abs \\ & = pr - pqs \\ a^{2n+1} + b^{2n+1} + c^{2n+1} & = p(r-qs) \end{align}

Therefore, $$a^{2n+1} + b^{2n+1} + c^{2n+1}$$ is divisible by $$p$$. $$\square$$ · 4 months, 1 week ago

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