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$2^3$ and $3^2$ are close together; $11^2$, $5^3$, and $2^7$ (121, 125, and 128) are close together; $3^5$, $2^8$, and maybe $17^2$ (243, 256, and 289) are close together. $7^3$ is close to $19^2$ (343 and 361). $3^7$ is very nearly $13^3$ (2187 and 2197), which is very nearly $47^2$ (2209). $19^4$ is close to $2^{17}$ (130,321 and 131,072). Further examples are easy to find.

One might expect coincidences to get farther apart and rarer as the numbers get larger, but that doesn't seem to happen. For example, $13^{11}$ and $23^9$ differ by about one part in 200 (1,792,160,394,037 and 1,801,152,661,463).

Is this all just the law of small numbers at work? How many such coincidences would one naïvely expect? Is there any evidence that the number of such coincidences is, or is not, more than one would naïvely expect? Is anything known about the distribution of prime powers?

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    Yes, that's the idea.2012-03-29

1 Answers 1

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For primes $p$ and $q$ the ratio $r=\log(p)/\log(q)$ is an irrational number and by the Equidistribution theorem the sequence $\{r,2r,3r,4r,\ldots\}$ is asymptotically equidistributed modulo 1.

Specifically, for large $N$ we would expect that $2\epsilon N$ elements of $\{i\cdot r\}_{i=1}^N$ are equivalent mod 1 to a number in the intervals $[0,\epsilon]\cap[1-\epsilon,1)$. This implies that $p^i$ is within a factor of $q^\epsilon$ of a power of $q$.

So if we look at powers of $p$ up to $N$ and want one to be within about 1 part in 200 of a power of $q$, approximately we want $\left|\log\left(p^a/q^b\right)\right|\le 0.005$, then we would expect to find $2N\log(1.005)/\log(q)$ close pairs.

Below are some charts showing this estimate and the actual number of pairs of exponents that give powers within 0.005 for pairs of primes $2\le p. The x-axis enumerates the prime pairs, starting at $(2,3)$ and ending at $(23,29)$. The 17th entry showing a count of 2 in the first chart is $(3,17)$.

Close pairs with N=100 Close pairs with N=1000 Close pairs with N=10000

Note that the above argument does not rely on $p,q$ being primes, only that $\log(p)/\log(q)$ is irrational. Here are tables for $N=100$ and $N=10000$ for a few sets of primes as well as $(2,\pi)$ and $(\zeta(3),\mathrm{e})$.

$ \begin{array}{|c|c|ccc|} p & q & Best & |\log| & \#\{|\log|\le 0.005\} & Expected \\ \hline 2 & 3 & 2^{84} \sim 3^{53} & 0.0021 & 1 & 0.9 \\ 2 & 5 & 2^{65} \sim 5^{28} & 0.0097 & 0 & 0.6 \\ 5 & 13 & 5^{51} \sim 13^{32} & 0.0030 & 1 & 0.4 \\ 3 & 17 & 3^{49} \sim 17^{19} & 0.0009 & 2 & 0.4 \\ 13 & 17 & 13^{95} \sim 17^{86} & 0.0138 & 0 & 0.4 \\ 11 & 23 & 11^{17} \sim 23^{13} & 0.0028 & 1 & 0.3 \\ 17 & 29 & 17^{82} \sim 29^{69} & 0.0199 & 0 & 0.3 \\ 1229 & 1381 & 1229^{62} \sim 1381^{61} & 0.0009 & 1 & 0.1 \\ 2 & \pi & 2^{71} \sim \pi^{43} & 0.0099 & 0 & 0.9 \\ \zeta(3) & \mathrm{e} & \zeta(3)^{38} \sim \mathrm{e}^{7} & 0.0067 & 0 & 1.0 \\ \hline \end{array} $

$ \begin{array}{|c|c|ccc|} p & q & Best & |\log| & \#\{|\log|\le 0.005\} & Expected \\ \hline 2 & 3 & 2^{1054} \sim 3^{665} & 0.00004 & 90 & 90.8 \\ 2 & 5 & 2^{9297} \sim 5^{4004} & 0.00006 & 62 & 62.0 \\ 5 & 13 & 5^{9551} \sim 13^{5993} & 0.000002 & 38 & 38.9 \\ 3 & 17 & 3^{5965} \sim 17^{2313} & 0.00016 & 37 & 35.2 \\ 13 & 17 & 13^{1637} \sim 17^{1482} & 0.00008 & 34 & 35.2 \\ 11 & 23 & 11^{4489} \sim 23^{3433} & 0.00024 & 30 & 31.8 \\ 17 & 29 & 17^{2875} \sim 29^{2419} & 0.00025 & 28 & 29.6 \\ 1229 & 1381 & 1229^{7813} \sim 1381^{7687} & 0.00012 & 16 & 13.8 \\ 2 & \pi & 2^{9217} \sim \pi^{5581} & 0.00007 & 86 & 87.1 \\ \zeta(3) & \mathrm{e} & \zeta(3)^{1641} \sim \mathrm{e}^{302} & 0.00008 & 98 & 99.8 \\ \hline \end{array} $

It doesn't seem that the number of close pairs is more than expected. For $N=100$ and a 0.005 cutoff the average count is slightly less than we might expect from the asymptotics, but by $N=10000$ the observed match the model quite closely.

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    Thanks again for the effort you put into this answer. I am very grateful.2012-05-11