38
$\begingroup$

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.

Improve this question
$\endgroup$
17
  • 6
    $\begingroup$ Besides the excellent question, what are other almost integers that have an "explanation" except $e^{\pi\sqrt{163}}$ and similar numbers? $\endgroup$
    – Gjergji Zaimi
    Jun 12, 2010 at 14:33
  • 18
    $\begingroup$ "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
    Jun 12, 2010 at 14:53
  • 2
    $\begingroup$ 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
    Jun 12, 2010 at 15:23
  • 12
    $\begingroup$ $\| n / log(n) \|$ (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
    Jun 12, 2010 at 16:00
  • 3
    $\begingroup$ 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
    Jun 12, 2010 at 16:54

2 Answers 2

Reset to default
4
$\begingroup$

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

Improve this answer
$\endgroup$
11
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.