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The j-function and the fact that 163 and 67 have class number 1 explain why: Exp(Pi*Sqrt(163)) = 262537412640768743.99999999999925, Exp(Pi*Sqrt(67)) = 147197952743.9999987. But is there any explanation for these?: 163/ln(163) = 31.9999987 $\approx $ 2^5, 67/ln(67) = 15.93 $\approx $ 2^4, 17/ln(17) = 6.00025. These numbers seem too close to integers to occur by chance. |
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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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