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answered  Exponential Sum Bound 
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
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Dynamics in the integers  Floor function
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Dynamics in the integers  Floor function
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Dynamics in the integers  Floor function
@Lucia: I just wrote the same. But you probably beat me by a minute or so. 
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Dynamics in the integers  Floor function
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answered  Dynamics in the integers  Floor function 
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What does a Turing machine compute?
See also en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis 
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awarded  Nice Answer 
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
@KierenMacMillan: Yes, this is why I kept the previous example of $0.485$ as well. 
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
@Joël: Thanks for your comment. I will update the text accordingly. 
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
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Counterexample to Pólya's conjecture
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
Consider the odd pairs $(\ell,m)$ satisfying $\ell^2\mid m^32$. Then the ratios $m/\ell$ are equidistributed in $(0,\infty)$ in a certain sense (I can be more specific if you wish). In particular, the ratios $m/\ell$ are dense in $(0,\infty)$, hence they are infinitely often smaller than $1/3$. 
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
@KierenMacMillan: One counterexample does not tell much, beyond disproving your conjecture of course. Infinitely many counterexamples would tell that the abc conjecture cannot be strengthened in a certain way. At any rate, here is a conjecture that I believe is true and would yield infinitely many counterexamples to your conjecture (continued in next comment): 
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Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^21)$
We are now at $m/\ell\approx 0.485$ which is smaller than all previous ratios, including the one for $(\ell,m)=(5,3)$. 