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The $j$-function and the fact that 163 and 67 have class number 1 explain why:

$\operatorname{exp}(\pi\cdot \sqrt{163}) = 262537412640768743.99999999999925$,

$\operatorname{exp}(\pi\cdot \sqrt{67}) = 147197952743.9999987$.

But is there any explanation for these?:

$\frac{163}{\operatorname{ln}(163)} = 31.9999987 \approx 2^5$,

$\frac{67}{\operatorname{ln}(67)} = 15.93 \approx 2^4$,

$\frac{17}{\operatorname{ln}(17)} = 6.00025$.

These numbers seem too close to integers to occur by chance.

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    $\begingroup$ Besides the excellent question, what are other almost integers that have an "explanation" except $e^{\pi\sqrt{163}}$ and similar numbers? $\endgroup$
    – Gjergji Zaimi
    Commented Jun 12, 2010 at 14:33
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    $\begingroup$ "These numbers seem too close to integers to occur by chance" Why that? The set of near-integers is simply to big to have a reasonable explanation for every single element. If 15.9 is a near-integer for you, then it seems that ~20% of all real numbers (i.e. all elements of $[0,1/10) \cup (9/10,1]$ and all its translates) are near-integers. That's way to much to "explain" anything. Most of these numbers are near-integers because of pure probability. $\endgroup$
    – Johannes Hahn
    Commented Jun 12, 2010 at 14:53
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    $\begingroup$ Max, a very nice explanation of the almost integrality of $e^{\pi\sqrt{-D}}$ can be found in Zagier's lectures "Elliptic modular forms and their applications", Chapter 6 (published recently by Springer in "1-2-3 modular forms"). He also explains a high factorisation properties of the corresponding integers (they are cubes!). $\endgroup$
    – Wadim Zudilin
    Commented Jun 12, 2010 at 15:23
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    $\begingroup$ $\| n / log(n) \|$ (distance to the nearest integer) hits a new record low for $n=$ 2, 5, 9, 13, 17, 163, 53453, 110673, 715533, .... $\endgroup$
    – Kevin O'Bryant
    Commented Jun 12, 2010 at 16:00
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    $\begingroup$ Here's a question: is this an exceptional number of record lows? I'd also argue that the fact that 53453/163 is exceptionally large (naively one would expect each record low to be something like $e$ times the previous one) is connected to the fact that 163/log(163) is exceptionally near an integer. (If we picked real numbers at random, only one out of every 400,000 is as close to an integer as 163/log(163).) Compare, say, 355/113 as an approximation of $\pi$. $\endgroup$
    – Michael Lugo
    Commented Jun 12, 2010 at 16:54

2 Answers 2

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On the other hand, Mathematica gives LogIntegral[163]=43.075210908806756346563... and LogIntegral[67]=22.6520420103880266691324... so this does not appear to be connected to x/Ln[x] in the context of the Prime Number Theorem

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What make us confident about some mystery in these observations?

1st note: "An example discovered by Srinivasa Ramanujan around 1913 is $\exp(\pi\sqrt{163})$, which is an integer to one part in $10^{30}$, and has second continued fraction term $1,333,462,407,511$. (This particular example can be understood from the fact that as $d$ increases $\exp(\pi\sqrt{d})$ becomes extremely close to $j((1 + \sqrt{-d})/2)$, which turns out to be an integer whenever there is unique factorization of numbers of the form $a + b \sqrt{-d}$ --- and $d=163$ is the largest of the 9 cases for which this is so.) Other less spectacular examples include $e^{\pi}-\pi$ and $163/\log(163)$."

2nd note: "Any computation involving 163 gives an answer that is close to an integer: $$ 163\pi = 512.07960\dots, \quad 163e = 443.07993\dots, \quad 163\gamma = 94.08615\dots\text{"} $$ and $$ \text{"}67/\log(67)=15.9345774031\dots, \quad 43/\log(43)=11.432521184\dots $$ ...nah, with class number 1 it's not connected. It's just the same 163. $\ddot\smile$"

A synthetic example of my own: $$ \root3\of{163}-\frac{49,163}{9,000} =0.0000000157258\dots $$ (note the double appearance of 163).

So, let's feel that the prime 163 is a supernatural number. $\ddot\smile$

EDIT. Another interpretation the original question is related to the observation of Kevin O'Bryant who computed the first successive maxima of the sequence $\|n/\log(n)\|$ where $\|\ \cdot\ \|$ denotes the distance to the nearest integer. The existence of infinitely many terms is guaranteed by the following

Problem. For any $\epsilon>0$, there exists an $n$ such that $\|n/\log(n)\|<\epsilon$.

See solution by Kevin Ventullo to this question. I hope that this fact demystifies the original problem in full.

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