The well-ordering principle states that every non-empty set of positive integers contains a least element.
I need to prove that 9|$n^3+(n+1)^3+(n+2)^3$ , $n∈N$ using this principle.
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1Welcome to Math Stack Exchange. You could prove it using induction, since $n^3-(n-3)^3=9(n^2-3n+3)$ – J. W. Tanner Nov 10 '19 at 01:05
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WOP and induction are equivalent – J. W. Tanner Nov 10 '19 at 03:29
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If it is false then by WOP there is a least counterexample $k$ with $\,9\nmid f_k.\,$ Note $\,k>0\,$ by $\,f_0 = 9.\,$ Note $\bmod 9\!:\ f_{n+1}\!- f_n = (n\!+\!3)^3\!-n^3\equiv 0\,$ so $\,9\mid f_{n+1}\!\!\iff\! 9\mid f_n,\,$ so $\,\color{#c00}{9\nmid f_k\Rightarrow 9\nmid f_{k-1}},\,$ so $\,k\!-\!1\ge 0\,$ is a smaller counterexample, contradiction.
Remark $ $ It is more natural to use ascent (induction) than $\rm\color{#c00}{descent}$ (WOP). Indeed, the proof shows that, modulo $9,\,$ the value of $f_n$ never changes $\,f_{n+1}\equiv f_n\,$ so reformulating the above positively as an induction proves that $\,f_n\,$ is a constant sequence $\,f_n \equiv f_0\,$ for all $n\,$ (and here $\,f_0 = 9\equiv 0)$.
Bill Dubuque
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Hi, thanks for your answer. Could you please explain this step: $(n+3)^3−n^3≡0$ ? – Jewgah Nov 10 '19 at 08:29
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@The Binomial Theorem $\large \Rightarrow (n!+!\color{#c00}{3})^{\large\color{#0a0}3} = 3^{\large 0} n^{\large 3} + \color{#0a0}{3,\cdot},\color{#c00}{ 3^{\large 1}} n^2 + \color{#c00}{3^{\large 2}}\color{}{(\cdots)}\equiv n^{\large 3}\color{#0a0}{\pmod {!\color{#c00}9}},\ $ i.e. all summands (except first $\large ,= n^3),$ have coef's divisible by $,\color{#c00}9,$ so they all are $\large\equiv 0\pmod{!\color{#c00}{9}}\ \ \ $ – Bill Dubuque Nov 10 '19 at 14:55
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@The proof does not claim $, 9\mid n^3.\ $ Where did that come from? – Bill Dubuque Nov 10 '19 at 15:52
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My bad, I misunderstood because of the $n^3(mod9)$ and $≡0(mod9)$. Thanks ! – Jewgah Nov 10 '19 at 16:23
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Direct proof: $$n^3+(n+1)^3+(n+2)^3\equiv \\ 3(n+1)(n^2+2n+3)\equiv \\ 6n(n+1)(n+2)+9(n+1) \equiv 0 \pmod{9}.$$ Can you interpret why the first addend is a multiple of $9$?
farruhota
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