MHB Find Integer Pairs $x,y$ with Infinitely Many Solutions

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The discussion focuses on finding integer pairs \(x, y > 3\) such that there are infinitely many positive integers \(k\) making the expression \(\frac{k^x+k-1}{k^y+k^2-1}\) an integer. Participants mention that the pairs of integers satisfying the condition include \((x, y) = (5, 3)\). There is a clarification that the problem may actually allow for the inequality \(x, y \ge 3\). The conversation highlights the importance of precise conditions for the problem. Overall, the thread emphasizes the exploration of integer solutions in the context of the given mathematical expression.
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Find all pairs of integers $x,\,y>3$ such that there exist infinitely many positive integers $k$ for which $\dfrac{k^x+k-1}{k^y+k^2-1}$ is an integer.
 
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Solution of other:

The problem is saying find all pairs of integers $x,\,y>3$ for which $k^y+k^2-1$ is a factor of the polynomial $k^x+k-1$.

It's clear that $x>y$. Let $x=y+a$. We have

$k^x+k-1=k^a(k^y+k^2-1)+(1-k)(k^{a+1}+k^a-1)$

So $k^y+k^2-1$ divides $k^{a+1}+k^a-1$. Now, $k^y+k^2-1$ has a real root $\alpha\in(0,\,1)$, $\alpha$ is also a root of $k^{a+1}+k^a-1$. Thus, $\alpha^{a+1}+\alpha^a=1$ and $\alpha^{y}+\alpha^2=1$. But $a+1>y>3$ and so $\alpha^{y}+\alpha^2>\alpha^{a+1}+\alpha^a$ with equality iff $a+1=y$ and $a=2$. Thus, $(x,\,y)=(5,\,3)$ is the only possible solution.
 
anemone said:
Solution of other:

The problem is saying find all pairs of integers $x,\,y>3$ for which $k^y+k^2-1$ is a factor of the polynomial $k^x+k-1$.

It's clear that $x>y$. Let $x=y+a$. We have

$k^x+k-1=k^a(k^y+k^2-1)+(1-k)(k^{a+1}+k^a-1)$

So $k^y+k^2-1$ divides $k^{a+1}+k^a-1$. Now, $k^y+k^2-1$ has a real root $\alpha\in(0,\,1)$, $\alpha$ is also a root of $k^{a+1}+k^a-1$. Thus, $\alpha^{a+1}+\alpha^a=1$ and $\alpha^{y}+\alpha^2=1$. But $a+1>y>3$ and so $\alpha^{y}+\alpha^2>\alpha^{a+1}+\alpha^a$ with equality iff $a+1=y$ and $a=2$. Thus, $(x,\,y)=(5,\,3)$ is the only possible solution.
the pairs of x,y>3 are given
but your solution: (x,y)=(5,3)
 
Last edited:
Albert said:
the pairs of x,y>3 are given
but your solution: (x,y)=(5,3)

Ops!:o Seems to me this problem is set for the loose inequality $x,\,y\ge 3$, thank you very much Albert for pointing this out! I appreciate it!(Nod)

Sorry for the late reply too...:(
 
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