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JOMO 6, Long 1

Find, with proof, all values of \(x\in\mathbb{N}\) such that \(\frac{x^x+1}{x+1}\) is a natural number.

Note by Yan Yau Cheng
2 years, 10 months ago

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sorry, it would be so if there would be -1 instead of +1 Bhavya Jain · 2 years, 4 months ago

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It can be expressed as a sum of a GP for all values of x belong to natural number. Bhavya Jain · 2 years, 4 months ago

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For uneven values of \(y\), the expression \(\dfrac{x^y+1}{x+1}\) is always a natural number. Having now \(x=y\), we have that our desired expression \(\dfrac{x^x+1}{x+1}\) is a natural number for \(\boxed{x = 2k + 1, k \in \mathbb{N}.}\) Guilherme Dela Corte · 2 years, 10 months ago

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@Guilherme Dela Corte There is one flaw in your proof: you did not prove that the number \(\dfrac{x^x+1}{x+1}\) is not an integer or all even \(x\).

Thankfully, this is pretty straightforward to do. Daniel Liu · 2 years, 9 months ago

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@Guilherme Dela Corte Could you explain your first assumption? Finn Hulse · 2 years, 10 months ago

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@Finn Hulse Finn, please note that \(\dfrac{a^{2k+1} + b^{2k+1} }{a+b} = a^{2k} + a^{2k-1} b + a^{2k-2} b^2 + \cdots + a^2 b^{2k-2} + a b^{2k-1} +b^{2k}\) (you can quickly check this by trying some small cases).

Since both \(a,b\) are integers, we have then that \(\dfrac{a^c + b^c }{a+b}\) is an integer for odd = uneven values of \(c\). Now let \(a=c\) and \( b=1\).

Note: math fixed by mod. Guilherme Dela Corte · 2 years, 10 months ago

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@Guilherme Dela Corte Guilherme, I would prefer a proof rather than

"(you can quickly check this by trying some small cases)"

please. This is the assumption I am referring to. Finn Hulse · 2 years, 10 months ago

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@Finn Hulse The proof is right there: just clear denominators of

\(\dfrac{a^{2k+1} + b^{2k+1} }{a+b} = a^{2k} + a^{2k-1} b + a^{2k-2} b^2 + \cdots + a^2 b^{2k-2} + a b^{2k-1} +b^{2k}\)

and you have proved it. Daniel Liu · 2 years, 9 months ago

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@Daniel Liu Oh shoot. I was just skimming, it didn't seem like much. Also dude do you play League? Finn Hulse · 2 years, 9 months ago

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For a positive integer x,n (where n=x+1), L=(1+x^x)/(1+x)= [1+(n-1)^(n-1)]/[n]. By remainder theorem, (n-1)^(n-1)=n(m)+(-1)^(n-1), for some integer m. L=[n(m)+((-1)^(n-1)+1)]/n. Since L is integer, (-1)^(n-1)+1 should be 0. That means, n-1 should be a odd number. Therefore, x=n-1 should be a odd number. ~ 汶汶 樂 · 2 years, 10 months ago

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