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MathJax: \mid for divisibility
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Martin Sleziak
Martin Sleziak
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Some equations similar to goormaghtighGoormaghtigh problem

Let $m,n,t$ be some primes and $x$ and $y$ be some prime powers. Also suppose that $mn|(t-1)$$mn\mid(t-1)$. Which one of the following is correct?

1- If $t=\dfrac{x^{n}-1}{(x-1)(x-1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then $x=y$ and $m=n$.

2- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then there is no solution for $x,y,m,n$.

3- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}+1}{(y+1)(y+1,m)}$, then $x=y$ and $m=n$.

Some equations similar to goormaghtigh problem

Let $m,n,t$ be some primes and $x$ and $y$ be some prime powers. Also suppose that $mn|(t-1)$. Which one of the following is correct?

1- If $t=\dfrac{x^{n}-1}{(x-1)(x-1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then $x=y$ and $m=n$.

2- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then there is no solution for $x,y,m,n$.

3- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}+1}{(y+1)(y+1,m)}$, then $x=y$ and $m=n$.

Some equations similar to Goormaghtigh problem

Let $m,n,t$ be some primes and $x$ and $y$ be some prime powers. Also suppose that $mn\mid(t-1)$. Which one of the following is correct?

1- If $t=\dfrac{x^{n}-1}{(x-1)(x-1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then $x=y$ and $m=n$.

2- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then there is no solution for $x,y,m,n$.

3- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}+1}{(y+1)(y+1,m)}$, then $x=y$ and $m=n$.

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Zahra Mombeyni
Zahra Mombeyni
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Some equations similar to goormaghtigh problem

Let $m,n,t$ be some primes and $x$ and $y$ be some prime powers. Also suppose that $mn|(t-1)$. Which one of the following is correct?

1- If $t=\dfrac{x^{n}-1}{(x-1)(x-1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then $x=y$ and $m=n$.

2- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then there is no solution for $x,y,m,n$.

3- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}+1}{(y+1)(y+1,m)}$, then $x=y$ and $m=n$.