| Well, its about 163...
Actually, I'd just expand LN(X+DX)^2 in a Taylor series about 640320**3
with DX equal to 744. With X0=640320 the series would be
LN(X0^3+DX)^2 =
9*LN(X0)^2 + 6*LN(X0)*DX/(X0^3) + (1-3*LN(X0))*DX^2/(X0^6) + ...
With a relative change of about 1.4E-16 due to the first order term.
So a zero order approx is below 163 by about 2.13E-14 and a first order
approx is above by about 3.55E-15 and differs insignificantly from
higher order approximations.
FORTRAN REAL*8 precision is adequate for all this...
- Jim
|
| On second thought, you would need a bit more precision than REAL*8...
Replacing 744 with 743.999999999999246 gives an even closer approx to
163. Pretty surprising. You could probably represent the numerator and
denominator as continued fractions and show how most terms cancel leaving a
very close approx to an integer result, who knows...
- Jim
|
| This might be of some interest on this surprising near-integer
expression...
- Jim
Newsgroups: sci.math
Path: decwrl!labrea!rutgers!tut.cis.ohio-state.edu!bgsuvax!steiner
Subject: explanation of some apparently amazing phenomena wanted
Posted: 25 Jan 89 19:56:01 GMT
Organization: Bowling Green State University B.G., Oh.
Here are two expressions that are very nearly integers:
a) exp(pi*sqrt(163))= 262537412640768743.99999999999925007...
b)[(1/pi)*(log(640320**3+744)]**2= 163.0000000000000000000000000000232...
Both these phenomena are related to the fact that the class
number of q(sqrt(-163))=1 and that the value of a certain
modular function j(z) at 163(and indeed at other values of
d for which the class number of q(sqrt(-d))=1 ) is very
close to an integer. My question is: There was a recently
published article on such phenomena that explains such
things. Does anyone know in which journal(and which issue)
this article appeared?
Also, how is the expression on the left side of b derived?
640320 is the value of gamma((-3+sqrt(-163))/2) while 744
is the constant of the z term in the power series expansion
of j(z). Could someone make this more precise? Thanks
Ray Steiner@BGSUVAX
Newsgroups: sci.math
Path: decwrl!Angelo!labrea!polya!Gang-of-Four!ilan
Subject: Re: explanation of some apparently amazing phenomena wanted
Posted: 25 Jan 89 23:39:15 GMT
Organization: Computer Science Department, Stanford University
In article <[email protected]> [email protected] (Ray Steiner) writes:
>Here are two expressions that are very nearly integers:
>a) exp(pi*sqrt(163))= 262537412640768743.99999999999925007...
>b)[(1/pi)*(log(640320**3+744)]**2= 163.0000000000000000000000000000232...
>Both these phenomena are related to the fact that the class
>number of q(sqrt(-163))=1 and that the value of a certain
>modular function j(z) at 163(and indeed at other values of
>d for which the class number of q(sqrt(-d))=1 ) is very
>close to an integer.
The theory of complex multiplication and ``Kronecker's Jugendtraum''
say that j(a+ b sqrt{-d}), d>0, will be an algebraic integer of degree
the class number of Q(sqrt{-d}). On the other hand,
j(z) = e^{- 2pi i z} + 744 + O( e^{2pi i z})
so the O( ) is exponentially small as d => infinity. Therefore
j(a+b sqrt{-d}) gets very close to algebraic integers.
The 163 example is the special case of
j( 1/2 + sqrt{-163} / 2)
and the class number for sqrt{-163} is one as observed.
The only reference that I know that gives the whole ``Jugendtraum''
is ``Seminar in Complex Multiplication'' by Chowla, Serre, Borel, Herz,...
which is a very early Springer Lecture Notes. Stark mentions sqrt{163}
and the formula above and its relation to the fact that x^2-x+41
represents primes for x=0,...,40 (note that it has discriminant -163)
and continued fractions in an article published in the AMS symposia in
Pure Mathematics around 1970.
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