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I want to prove that there are only finitely many rational solutions to $$\left|\frac{\log(5)}{\log(7)}-\frac{a}{b}\right|\le \frac{1}{7^b}$$

And once I have done this, I would like to put a bound on how many solutions there are. Any thoughts?

Edit:

Since for any $x$ and large $b$, $7^b>b^x$, if my equation has infinitely many solutions, then $$\frac{\log(5)}{\log(7)}$$ Must have infinite irrationality measure. Can we prove this isn't true?

beep-boop
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Elie Bergman
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    what is the source of the problem? And, what is your background as relates to this one? – Will Jagy Jul 28 '14 at 19:16
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    What do you mean by rational solutions? You mean that $a,b\in\mathbb Q$ or $a,b\in\mathbb N$? – Yiorgos S. Smyrlis Jul 28 '14 at 20:09
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    @YiorgosS.Smyrlis, these will be convergents of the simple continued fraction for the irrational target number given (that is $a,b$ positive integers). There probably are none satisfying the inequality, which is very strong. – Will Jagy Jul 28 '14 at 20:23
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    Chances are, your number has irrationality measure 2 --- almost all number do --- but I doubt there's a proof known for your number. – Gerry Myerson Jul 29 '14 at 09:12

2 Answers2

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Assuming $a,b$ are integers, it can be verified by brute-force search that if there are any solutions, then $a\ge4000$ or $b\ge2000$.

Checked with Mathematica with following code:

Monitor[Do[If[Abs[Log[5]/Log[7] - a/b] <= 1/7^b, Print[{a, b}]], {b, 1, 1999}, {a, 1, 3999}], {b, a}]

Using Will Jagy's advice, checking continued fractions up to 19 terms no solutions found (more than 19 terms Mathematica overflows).

Do[t = FromContinuedFraction@ContinuedFraction[Log@5/Log@7, a]; If[Abs[Log@5/Log@7 - t] <= 1/7^Denominator@t, Print@a], {a, 1, 19}]
Somnium
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    Theorem 19 in Khinchin's little book, if the absolute value is less than $1/(2 b^2)$ for coprime $a,b,$ then $a/b$ must be a continued fraction convergent of the irrational, here $\alpha = \frac{\log 5}{\log 7}$ – Will Jagy Jul 28 '14 at 20:19
  • So to prove there are no solutions we must examine the CF? How could one prove that no convergent satisfies our condition? – Elie Bergman Jul 29 '14 at 08:37
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I believe Bakers theorem can put a bound on the number of solutions to this equation.

Elie Bergman
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