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David Roberts
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Algorihtm Algorithm for least distance of powers of integers

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From Michailescu's theorem (Catalan's conjecture) we have that the only $a,b,m,n \in \mathcal{Z}^{+}$ with $m,n>1$ such that $a^{m} - b^{n} = 1$ are: $a=3$, $b=2$, $m=2$, $n=3$.

  1. Is there an algorithm which, for any $a,b \in \mathcal{Z}^{+}$ finds the minimum of $|a^{m} - b^{n}|$ $\forall m,n \in \mathcal{Z}^{+}$ (with $m,n>1$ or $m,n>0$)?

  2. Do we know for how many different values of $m,n$ this minimum distance can be achieved?

  3. If we do not have such an algorithm, do we know if this problem is decidable?

From Michailescu's theorem (Catalan's conjecture) we have that the only $a,b,m,n \in \mathcal{Z}^{+}$ with $m,n>1$ such that $a^{m} - b^{n} = 1$ are: $a=3$, $b=2$, $m=2$, $n=3$.

  1. Is there an algorithm which, for any $a,b \in \mathcal{Z}^{+}$ finds the minimum of $|a^{m} - b^{n}|$ $\forall m,n \in \mathcal{Z}^{+}$ (with $m,n>1$ or $m,n>0$)?

  2. Do we know for how many different values of $m,n$ this minimum distance can be achieved?

  3. If we do not have such an algorithm, do we know if this problem is decidable?

From Michailescu's theorem (Catalan's conjecture) we have that the only $a,b,m,n \in \mathcal{Z}^{+}$ with $m,n>1$ such that $a^{m} - b^{n} = 1$ are: $a=3$, $b=2$, $m=2$, $n=3$.

  1. Is there an algorithm which, for any $a,b \in \mathcal{Z}^{+}$ finds the minimum of $|a^{m} - b^{n}|$ $\forall m,n \in \mathcal{Z}^{+}$ (with $m,n>1$ )?

  2. Do we know for how many different values of $m,n$ this minimum distance can be achieved?

  3. If we do not have such an algorithm, do we know if this problem is decidable?

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From Michailescu's theorem (Catalan's conjecture) we have that the only $a,b,m,n \in \mathcal{Z}^{+}$ with $m,n>1$ such that $a^{m} - b^{n} = 1$ are: $a=3$, $b=2$, $m=2$, $n=3$.

  1. Is there an algorithm which, for any $a,b \in \mathcal{Z}^{+}$ finds the minimum of $|a^{m} - b^{n}|$ $\forall m,n \in \mathcal{Z}^{+}$ (with $m,n>1$ or $m,n>0$)?

  2. Do we know for how many different values of $m,n$ this minimum distance can be achieved?

  3. If we do not have such an algorithm, do we know if this problem is decidable?

From Michailescu's theorem (Catalan's conjecture) we have that the only $a,b,m,n \in \mathcal{Z}^{+}$ with $m,n>1$ such that $a^{m} - b^{n} = 1$ are: $a=3$, $b=2$, $m=2$, $n=3$.

  1. Is there an algorithm which, for any $a,b \in \mathcal{Z}^{+}$ finds the minimum of $|a^{m} - b^{n}|$ $\forall m,n \in \mathcal{Z}^{+}$ (with $m,n>1$)?

  2. Do we know for how many different values of $m,n$ this minimum distance can be achieved?

  3. If we do not have such an algorithm, do we know if this problem is decidable?

From Michailescu's theorem (Catalan's conjecture) we have that the only $a,b,m,n \in \mathcal{Z}^{+}$ with $m,n>1$ such that $a^{m} - b^{n} = 1$ are: $a=3$, $b=2$, $m=2$, $n=3$.

  1. Is there an algorithm which, for any $a,b \in \mathcal{Z}^{+}$ finds the minimum of $|a^{m} - b^{n}|$ $\forall m,n \in \mathcal{Z}^{+}$ (with $m,n>1$ or $m,n>0$)?

  2. Do we know for how many different values of $m,n$ this minimum distance can be achieved?

  3. If we do not have such an algorithm, do we know if this problem is decidable?

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