More on e^(pi*sqrt(163)) - sci.math.research

Message de la discussion More on e^(pi*sqrt(163))

Alexander R. Povolotsky

Afficher le profil

Autres options 27 juin 2009, 10:08

Groupes de discussion : sci.math.research

De : Alexander R.Povolotsky <apovo...@gmail.com>

Date : Sat, 27 Jun 2009 15:08:21 +0100 (BST)

Date/heure locale : Sam 27 juin 2009 10:08

Objet : Re: More on e^(pi*sqrt(163))

  Transférer | Imprimer | Afficher le fil
  | Afficher l'original
  | Signaler ce message | Rechercher les messages de cet auteur
  

I suggest to consider following for above, namely:
the last four of Class Number 1 expressions in
http://www.geocities.com/titus_piezas/Ramanujan_a.htm
could be generalized as:

exp(Pi*sqrt(19+24*n) =~ (24*k)^3 + 31*24

PARI confirmation is below ..

gp > for(n=0,10,print1("n= ",n," k= ",
((exp(Pi*sqrt(19+24*n))/24-31)/24/24)^(1/3),"\n"))

n= 0 k= 3.999999664954872711861691865 <<========
n= 1 k= 39.99999999999664632214064072 <<========
n= 2 k= 219.9999999999999993336409313 <<========
n= 3 k= 908.2994607084626509324663895
n= 4 k= 3139.719720204852366879238790
n= 5 k= 9587.574481226312121129336932
n= 6 k= 26680.00000000000000000000000 <<========
n= 7 k= 69020.39408641981880200520050
n= 8 k= 168277.4270764306998213353795
n= 9 k= 390498.9836593266367110562264
n= 10 k= 868910.8509221483459190206684

Cheers,
Alexander R. Povolotsky

---------- Forwarded message ----------
From: tpie...@gmail.com
Date: Apr 13 2008, 1:28 am
Subject: More on e^(pi*sqrt(163))
To: sci.math.research

Hello all,

It is quite well-known that:

e^(pi*sqrt(19)) ~ 96^3 + 744
e^(pi*sqrt(43)) ~ 960^3 + 744
e^(pi*sqrt(67)) ~ 5280^3 + 744
e^(pi*sqrt(163)) ~ 640320^3 + 744

using the four highest Heegner numbers. But it is not so well-known
that the expression e^(pi*sqrt(d)) can be given *another* internal
structure:

e^(pi*sqrt(19)) ~ 12^3(3^2-1)^3 + 744
e^(pi*sqrt(43)) ~ 12^3(9^2-1)^3 + 744
e^(pi*sqrt(67)) ~ 12^3(21^2-1)^3 + 744
e^(pi*sqrt(163)) ~ 12^3(231^2-1)^3 + 744

The reason for the squares are due to certain Eisenstein series -- but
that's another story. :-)

Beautifully consistent, aren't they?

I'm working on a new webpage about this and, er, other Ramanujan-
related stuff. But I'm having a devil of a time finishing it due to my
day job. I'll post the link here when it's done.

Yours,

Titus

Transférer