More on e^(pi*sqrt(163))

[tpi...@gmail.com]

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

[Alexander R.Povolotsky]

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

[Alexander Povolotsky]

As a follow-up to my previous posting re the "near integer" values of
k (obtained for n=0,1,2,6 ), it is interesting (IMHO) that we could
subtract (from those k)
3.<near one fractional part>
and observe that the subtraction result is dividable by 36

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

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 <<= - 3.9... 0
/ 36 = 0
n= 1 k= 39.99999999999664632214064072 <<= - 3.9... 36 /
36 = 1
n= 2 k= 219.9999999999999993336409313 <<= - 3.9... 216 /
36 = 6
n= 6 k= 26680.00000000000000000000000 <<= - 4 26676 / 36 = 741

Using PARI/GP the results of above described division by 36 could be obtained as
gp >b(n)=((exp(Pi*sqrt(19+24*n))/24-31)/24/24)^(1/3)
gp > for (n=0,3,print1((ceil(b((abs(n-1))!*n))-4)/36,"\n"))
0
1
6
741