(New page: =The Erdos-Woods Problem= ==Setup== '''Fundamental Theorem of Arithmetic''': Every integer $n >1$ can be expressed uniquely as a product $$n = p_1^{e_1} \cdots p_k^{e_k}$$ where the $p_i$...) |
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| − | =The | + | = The Erdös-Woods Problem= |
| − | + | The Erdös-Woods problem is a surprisingly far reaching question about integers. I has implications for mathematical logic, [[ABC triples]], [[elliptic curves]], and can be generalized to (affine) [[schemes]]. This generalization provides links between number theory and Nevanlinna's value distribution theory in complex analysis. Connections between Nevanlinna theory and number theory have already been emphasized by Lang and Vojta in connections with Roth's theorem. | |
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| − | + | This completely independent connection suggests that the ABC conjecture could stated for the ring of entire functions. I seems plausible that it could even be proven by one of our friendly neighborhood Nevanlinna theorists. | |
| − | ''' | + | '''UPDATE:''' Apparently ABC conjecture has already been proven for the field of meromorphic functions. This was the topic of a recent Séminaire Bourbaki, documented [http://arxiv.org/pdf/0811.3153v1 here]. |
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| − | + | = Algorithm = | |
| − | + | One way to think about the Erdös-Woods problem for integers is as follows. You are confronted by an Evil Wizard who tells you that he is thinking of an integer $n$. He will tell you the primes dividing $n$, $n+1$, and $n+2$ respectively. You are asked to determine $n$. Given $V(n)$ and $V(n+1)$ we can prove, by invoking a theorem of Shafarevich, that there are only finitely many possible values of $n$. | |
| − | + | A reasonable computational task is to compute this finite set. This involves computing the set of all elliptic curves with full two-torsion with good reduction outside a finite set of primes. Noam Elkies has suggested that the best way to do this might be to compute the set of all possible [[Lambda Invariants]] by solving an [[S-Unit Equation]]. These equations can be solved using Baker's method of [[Linear Forms in Logarithms]]. | |
Latest revision as of 16:48, 5 August 2010
The Erdös-Woods Problem
The Erdös-Woods problem is a surprisingly far reaching question about integers. I has implications for mathematical logic, ABC triples, elliptic curves, and can be generalized to (affine) schemes. This generalization provides links between number theory and Nevanlinna's value distribution theory in complex analysis. Connections between Nevanlinna theory and number theory have already been emphasized by Lang and Vojta in connections with Roth's theorem.
This completely independent connection suggests that the ABC conjecture could stated for the ring of entire functions. I seems plausible that it could even be proven by one of our friendly neighborhood Nevanlinna theorists.
UPDATE: Apparently ABC conjecture has already been proven for the field of meromorphic functions. This was the topic of a recent Séminaire Bourbaki, documented here.
Algorithm
One way to think about the Erdös-Woods problem for integers is as follows. You are confronted by an Evil Wizard who tells you that he is thinking of an integer $n$. He will tell you the primes dividing $n$, $n+1$, and $n+2$ respectively. You are asked to determine $n$. Given $V(n)$ and $V(n+1)$ we can prove, by invoking a theorem of Shafarevich, that there are only finitely many possible values of $n$.
A reasonable computational task is to compute this finite set. This involves computing the set of all elliptic curves with full two-torsion with good reduction outside a finite set of primes. Noam Elkies has suggested that the best way to do this might be to compute the set of all possible Lambda Invariants by solving an S-Unit Equation. These equations can be solved using Baker's method of Linear Forms in Logarithms.
