• I have obtained a curious aproximation to Apery constant Zeta(3)= 1.2020569031595942... (199/155)(16/165)^(3/2)Pi^3 Can someone test with a super computer
Message 1 of 9 , Mar 14 12:38 AM
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I have obtained a curious
aproximation to Apery constant Zeta(3)=
1.2020569031595942...

(199/155)(16/165)^(3/2)Pi^3

Can someone test with a super computer expressions of this type:

e=N[Zeta[3]/Pi^3,140]

F[a_,b_,c_,d_]:=N[e(c/d)(a/b)^1.5,140]

Do[x=F[a,b,c,d];If[Abs[x-1]<0.0000001,Print[a," ",b," ",c," ",d,"
",x]],{a,1,260},{b,1,260},{c,1,260},{d,1,260}]

For larger values ​​of the parameters of course.

Or other irrational expressions multiplied by Pi ^ 3

Sincerely

Sebastián Martín Ruiz

[Non-text portions of this message have been removed]
• ... Hi. To follow up on SMR s post and do some crude numerology, I computed Zeta(3)/Pi^3 to 1000 decimals:
Message 2 of 9 , Mar 14 9:59 AM
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--- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz <s_m_ruiz@...> wrote:
>
> I have obtained a curious
> aproximation to Apery constant Zeta(3)=
> 1.2020569031595942...
>
> (199/155)(16/165)^(3/2)Pi^3

Hi.
To follow up on SMR's post and do some crude numerology, I computed Zeta(3)/Pi^3 to 1000 decimals:
0.038768179602916798941119890318721149806234568039552579223126762123777137012286855271851...
and then computed its regular continued fraction expansion

[0; 25, 1, 3, 1, 6, 3, 1, 1, 2, 1, 10, 3, 2, 1, 19, 3, 2, 1, 1, 3, 2, 3, 5, 3, 1, 1, 7, 1, 1, 1, 2, 1, 364, 11, 1, 84, 9, 34, 1, 7, 1, 63, 7, 1, 1, 4, 1, 5, 4, 7, 1, 1, 1, 5, 1, 4, 1, 5, 5, 9, 1, 21, 1, 9, 1, 1, 3, 2, 7, 1, 8, 5, 1, 7, 5, 2, 3, 1, 1, 1, 1, 1, 1, 1, 7, 8, 2, 2, 2, 1, 2, 1, 2, 6, 33, 99, 1, 1, 14, 1, 7, 2, 1, 1, 3, 4, 7, 1, 6, 5, 3, 1, 7, 8, 3, 5, 2, 1, 104, 1, 1, 1, 3, 8, 5, 1, 2, 1, 1, 11, 7, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 8, 1, 1, 7, 2, 4, 1, 1, 11, 1, 1, 1, 1, 35, 7, 1, 2, 3, 6, 1, 3, 1, 5, 81, 1, 2, 2, 7, 94, 2, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 3, 1, 10, 75, 1, 2, 6, 3, 2, 7, 1, 1, 1, 6, 1, 4, 1, 1, 10, 1, 8, 2, 2, 1, 2, 1, 5, 6, 2, 1, 3, 1, 5, 1, 4, 3, 2, 5, 3, 3, 1, 10, 3, 1, 1, 9, 1, 1, 3, 3, 4, 1, 1, 2, 2, 1, 1, 15, 1, 15, 7, 4, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 8, 1, 1, 2, 1, 2, 4, 3, 1, 1, 3, 12, 2, 3, 1, 10, 2, 5, 19, 1, 4, 4, 3, 1, 3, 1, 6, 1, 7, 1, 4, 1, 1, 2, 7, 94, 7, 4, 2, 1, 4, 1, 1, 2, 1, 2, 11, 2, 2, 1, 24, 2, 2, 1, 1, 6, 1, 1, 5, 2, 2, 1, 1, 1, 10, 1, 19, 1, 3, 1, 1, 1, 6, 1, 161, 1, 7, 1, 4, 5, 1, 5, 1, 2, 1, 4, 12, 1, 6, 2, 1, 7, 1, 4, 108, 1, 5, 1, 1, 3, 4, 4, 1, 1, 7, 1, 32, 1, 1, 1, 1, 2, 2, 3, 1, 1, 4, 2, 1, 4, 14, 1, 1, 2, 3, 16, 1, 3, 1, 2, 1, 1, 5, 1, 15, 2, 1, 2, 30, 1, 94, 7, 1, 1, 1, 1, 1, 1, 2, 2, 6, 6, 4, 2, 1, 4, 1, 5, 19, 1, 1, 1, 4, 8, 22, 1, 2, 3, 1, 5, 1, 1, 5, 2, 10, 6, 3, 4, 13, 1, 1, 1, 2, 1, 1, 97, 1, 1, 5, 2, 8, 1, 1, 2, 1, 3, 36, 1, 1, 2, 2, 1, 40, 1, 13, 3, 1, 1, 226, 1, 5, 3, 1, 7, 3, 19, 1, 20, 49, 1, 1, 33, 1, 7, 2, 1, 4, 7, 3, 7, 8, 1, 1, 55, 1, 17, 6, 1, 1, 1, 2, 1, 1, 5, 2, 8, 26, 3, 5, 6, 1, 2, 2, 2, 3, 1, 2, 2, 1, 33, 1, 3, 2, 2, 42, 1, 1, 1, 3, 2, 2, 1, 26, 1, 4, 7, 13, 7, 1, 29, 11, 1, 2, 1, 4, 1, 14, 3, 5, 2, 59, 6, 2, 2, 1, 103, 1, 8, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 12, 3, 13, 2, 2, 1, 13, 1, 1, 40, 4, 1, 2, 3, 1, 3, 1, 1, 12, 1, 7, 1, 2, 1, 1, 49, 1, 2, 1, 3, 1, 3, 8, 1, 4, 1, 2, 5, 1, 3, 1, 7, 2, 2, 24, 3, 2, 3, 2, 11, 2, 1, 15, 1, 3, 54, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 5, 6, 2, 7, 2, 1, 1, 8, 1, 1, 5, 1, 2, 2, 4, 3, 8, 2, 1, 1, 8, 43, 1, 1, 1, 17, 1, 3, 4, 24, 1, 6, 2, 490, 6, 3, 1, 3, 8, 1, 1, 27, 2, 2, 1, 2, 1, 18, 5, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 14, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 4, 1, 3, 1, 4, 1, 6, 3, 1, 1, 1, 2, 2, 26, 1, 1, 1, 1, 1, 1, 29, 2, 2, 2, 21, 6, 1, 2, 4, 70, 1, 50, 2, 1, 2, 1, 1, 3, 1, 3, 4, 1, 3, 3, 4, 2, 48, 1, 2, 1, 16, 1, 2, 2, 1, 32, 329, 3, 4, 12, 1, 3, 10, 2, 2, 1, 1, 1, 1, 1, 3, 3, 7, 400, 1, 3, 7, 4, 2, 9, 2, 2, 41, 1, 2, 75, 1, 12, 4, 2, 47, 32, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 2, 1, 23, 1, 112, 12, 1, 2, 1, 11, 5, 5, 3, 7, 1, 1, 1, 8, 1, 12, 1, 8, 1, 5, 4, 9, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 4, 1, 2, 2, 1, 2, 38, 1, 2, 1, 2, 2, 38, 59, 1, 8, 4, 2, 1, 4, 2, 3, 1, 29, 19, 2, 369, 1, 1, 7, 1, 5, 9, 2, 3, 1, 1, 3, 2, 2, 34, 1, 1, 1, 11, 1, 3, 1, 8, 1, 10, 1, 1, 1, 32, 1, 3, 1, 1, 8, 4, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 8, 2, 3, 1, 15, 1, 2, 2, 1, 10, 5, 1, 2, 2, 4, 29, 8, 5, 1, 1, 2, 5, 1, 2, 1, 2, 6, 1, 1, 1, 11, 2, 5, 1, 2, 115, 6, 14, 4, 121, 1, 4, 2, 15, 34, 24, 6, 1, ...]

Now we ask, based on this, "is Sebastien Martin Ruiz on to something interesting?"

The regular continued fraction (RCF)
expansion of a number X is useful for detecting
if that number is rational -- which is equivalent to the RCF
terminating, for example 355/113=[3;7,16].

It also is useful for detecting if X is a quadratic irrational,
which is equivalent to the RCF being eventually periodic, for
example
sqrt2=[1;2,2,2,2,...]
and
15^(3/2) = [58; repeat(10, 1, 1, 4, 8, 12, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 12, 8, 4, 1, 1, 10, 116)].

Finally, it is useful for detecting if X is "unusual."
Almost all real numbers have random RCF partial quotients
sampled from the Gauss-Kuzmin distribution
http://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution
Any number which violently fails statistical tests for that
is "unusual." For example
e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14,...]

Here are the counts A of how many times N appeared as a partial quotient for N=1,2,...,100:

A=count for Zeta(3)/Pi^3 = 0.038768
B=corresponding count for Pi+2^(1/3)+7^(1/2) = 7.047265...
G=Gauss-Kuzmin expected count

N, A, B, G
1, 419, 402, 415.0374993
2, 174, 185, 169.9250014
3, 92, 94, 93.10940439
4, 50, 60, 58.89368905
5, 45, 36, 40.64198453
6, 27, 28, 29.74734345
7, 36, 21, 22.72007650
8, 26, 14, 17.92190798
9, 7, 14, 14.49956969
10, 10, 15, 11.97264165
11, 9, 10, 10.05366460
12, 9, 11, 8.562013557
13, 5, 8, 7.379530341
14, 5, 6, 6.426269095
15, 5, 9, 5.646563141
16, 2, 5, 5.000681040
17, 2, 6, 4.459648259
18, 1, 4, 4.001930553
19, 6, 7, 3.611253552
20, 1, 4, 3.275132038
21, 2, 5, 2.983858436
22, 1, 2, 2.729792802
23, 1, 1, 2.506855594
24, 3, 4, 2.310160687
25, 1, 1, 2.135744286
26, 3, 2, 1.980364109
27, 1, 1, 1.841346819
28, 0, 2, 1.716472527
29, 4, 3, 1.603885687
30, 1, 0, 1.502025127
31, 0, 0, 1.409570255
32, 4, 3, 1.325397401
33, 3, 2, 1.248546193
34, 2, 1, 1.178191153
35, 1, 0, 1.113620255
36, 1, 0, 1.054216386
37, 0, 0, 0.9994424311
38, 2, 1, 0.9488293993
39, 0, 1, 0.9019662943
40, 2, 1, 0.8584915734
41, 1, 2, 0.8180862026
42, 1, 2, 0.7804680132
43, 1, 1, 0.7453862058
44, 0, 0, 0.7126180220
45, 0, 0, 0.6819641182
46, 0, 1, 0.6532465388
47, 1, 0, 0.6263056815
48, 1, 0, 0.6009976942
49, 2, 1, 0.5771934627
50, 1, 0, 0.5547760091
51, 0, 0, 0.5336397688
52, 0, 1, 0.5136888564
53, 0, 0, 0.4948361984
54, 1, 0, 0.4770028103
55, 1, 0, 0.4601164965
56, 0, 1, 0.4441111278
57, 0, 0, 0.4289267856
58, 0, 0, 0.4145080281
59, 2, 0, 0.4008043244
60, 0, 0, 0.3877691866
61, 0, 0, 0.3753597376
62, 0, 0, 0.3635365655
63, 1, 0, 0.3522634356
64, 0, 1, 0.3415067120
65, 0, 0, 0.3312352143
66, 0, 0, 0.3214203598
67, 0, 0, 0.3120352996
68, 0, 0, 0.3030553501
69, 0, 1, 0.2944575600
70, 1, 0, 0.2862205664
71, 0, 0, 0.2783244500
72, 0, 0, 0.2707507354
73, 0, 1, 0.2634818134
74, 0, 0, 0.2565019512
75, 2, 1, 0.2497957052
76, 0, 1, 0.2433490748
77, 0, 1, 0.2371487818
78, 0, 0, 0.2311825580
79, 0, 0, 0.2254387130
80, 0, 0, 0.2199062779
81, 1, 0, 0.2145750055
82, 0, 0, 0.2094353705
83, 0, 0, 0.2044781363
84, 1, 0, 0.1996947880
85, 0, 0, 0.1950773880
86, 0, 1, 0.1906182871
87, 0, 0, 0.1863104142
88, 0, 0, 0.1821469860
89, 0, 0, 0.1781215085
90, 0, 0, 0.1742280648
91, 1, 0, 0.1704607381
92, 0, 0, 0.1668144771
93, 0, 0, 0.1632839428
94, 3, 0, 0.1598642283
95, 0, 0, 0.1565508601
96, 0, 0, 0.1533395088
97, 1, 1, 0.1502259897
98, 0, 0, 0.1472062620
99, 1, 0, 0.1442767180
100, 0, 0, 0.1414337502

"Chi Squared" values for departure of observed counts from Gauss-Kuzmin law:

for A: 560.1
for B: 536.4

I played around a bit more and found ChiSquared for several more "random numbers" like B, for example
Pi-2^(1/3)+7^(1/2)-exp(1)-ln(3)+erf(1) = 1.55322959...
has ChiSquared=771.9.
So Ruiz's number in (A) is within the typical range of ChiSquared.

In contrast, e has enormous ChiSquared=107491
or sqrt(7) has ChiSquared=189729, both easily detected in this
way as "unusual."

CONCLUSION:
Sorry, the first 1000 RCF quotients do not seem to indicate that
Zeta(3)/Pi^3 is in any way an unusual real number.

You all might want to try larger statistical analyses than this,
but that's what I'm seeing from the first 1000. (It might be good to create a fast "unusual number detector" software tool based on
the above techniques and maybe a Kolmogorov-Smirnov test.)
• I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3 was a low-degree low-height algebraic number. Result: If it has degree
Message 3 of 9 , Mar 14 10:46 AM
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I also attempted to use PSLQ to figure out whether
Zeta(3)/Pi^3
was a low-degree low-height algebraic number.
Result:
If it has degree<=10 then its height is at least 10^91.

So, sorry again:
Zeta(3)/Pi^3 again fails to look unusual.
• No magic here, just the concept that you can find approximations to basically anything if you have enough degrees of freedom. For instance:
Message 4 of 9 , Mar 14 11:53 AM
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No magic here, just the concept that you can find approximations to
basically anything if you have enough degrees of freedom.

For instance:

(83/779)(605/689)^(3/2)Pi^3 == 2.71828182845...

gives you the first 12 correct digits of e.

(121/634)(305/724)^(3/2)Pi^3 == 1.6180339887...

gives you the first 11 correct digits of the golden ratio (1+sqrt(5))/2.

Note that there's an even better approximation to Zeta(3) at:

(25/186)(86/197)^(3/2)Pi^3

On 3/14/2011 9:59 AM, WarrenS wrote:
>
>
> --- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz<s_m_ruiz@...> wrote:
>>
>> I have obtained a curious
>> aproximation to Apery constant Zeta(3)=
>> 1.2020569031595942...
>>
>> (199/155)(16/165)^(3/2)Pi^3
>
> Hi.
> To follow up on SMR's post and do some crude numerology, I computed Zeta(3)/Pi^3 to 1000 decimals:
> 0.038768179602916798941119890318721149806234568039552579223126762123777137012286855271851...
> and then computed its regular continued fraction expansion
>
> [0; 25, 1, 3, 1, 6, 3, 1, 1, 2, 1, 10, 3, 2, 1, 19, 3, 2, 1, 1, 3, 2, 3, 5, 3, 1, 1, 7, 1, 1, 1, 2, 1, 364, 11, 1, 84, 9, 34, 1, 7, 1, 63, 7, 1, 1, 4, 1, 5, 4, 7, 1, 1, 1, 5, 1, 4, 1, 5, 5, 9, 1, 21, 1, 9, 1, 1, 3, 2, 7, 1, 8, 5, 1, 7, 5, 2, 3, 1, 1, 1, 1, 1, 1, 1, 7, 8, 2, 2, 2, 1, 2, 1, 2, 6, 33, 99, 1, 1, 14, 1, 7, 2, 1, 1, 3, 4, 7, 1, 6, 5, 3, 1, 7, 8, 3, 5, 2, 1, 104, 1, 1, 1, 3, 8, 5, 1, 2, 1, 1, 11, 7, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 8, 1, 1, 7, 2, 4, 1, 1, 11, 1, 1, 1, 1, 35, 7, 1, 2, 3, 6, 1, 3, 1, 5, 81, 1, 2, 2, 7, 94, 2, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 3, 1, 10, 75, 1, 2, 6, 3, 2, 7, 1, 1, 1, 6, 1, 4, 1, 1, 10, 1, 8, 2, 2, 1, 2, 1, 5, 6, 2, 1, 3, 1, 5, 1, 4, 3, 2, 5, 3, 3, 1, 10, 3, 1, 1, 9, 1, 1, 3, 3, 4, 1, 1, 2, 2, 1, 1, 15, 1, 15, 7, 4, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 8, 1, 1, 2, 1, 2, 4, 3, 1, 1, 3, 12, 2, 3, 1, 10, 2, 5, 19, 1, 4, 4, 3, 1, 3, 1, 6, 1, 7, 1, 4, 1, 1, 2, 7, 94, 7, 4, 2, 1, 4, 1, 1, 2, 1, 2, 11, 2, 2, 1, 24, 2, 2, 1, 1, 6, 1, 1, 5, 2, 2, 1,
1, 1, 10, 1, 19, 1, 3, 1, 1, 1, 6, 1, 161, 1, 7, 1, 4, 5, 1, 5, 1, 2, 1, 4, 12, 1, 6, 2, 1, 7, 1, 4, 108, 1, 5, 1, 1, 3, 4, 4, 1, 1, 7, 1, 32, 1, 1, 1, 1, 2, 2, 3, 1, 1, 4, 2, 1, 4, 14, 1, 1, 2, 3, 16, 1, 3, 1, 2, 1, 1, 5, 1, 15, 2, 1, 2, 30, 1, 94, 7, 1, 1, 1, 1, 1, 1, 2, 2, 6, 6, 4, 2, 1, 4, 1, 5, 19, 1, 1, 1, 4, 8, 22, 1, 2, 3, 1, 5, 1, 1, 5, 2, 10, 6, 3, 4, 13, 1, 1, 1, 2, 1, 1, 97, 1, 1, 5, 2, 8, 1, 1, 2, 1, 3, 36, 1, 1, 2, 2, 1, 40, 1, 13, 3, 1, 1, 226, 1, 5, 3, 1, 7, 3, 19, 1, 20, 49, 1, 1, 33, 1, 7, 2, 1, 4, 7, 3, 7, 8, 1, 1, 55, 1, 17, 6, 1, 1, 1, 2, 1, 1, 5, 2, 8, 26, 3, 5, 6, 1, 2, 2, 2, 3, 1, 2, 2, 1, 33, 1, 3, 2, 2, 42, 1, 1, 1, 3, 2, 2, 1, 26, 1, 4, 7, 13, 7, 1, 29, 11, 1, 2, 1, 4, 1, 14, 3, 5, 2, 59, 6, 2, 2, 1, 103, 1, 8, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 12, 3, 13, 2, 2, 1, 13, 1, 1, 40, 4, 1, 2, 3, 1, 3, 1, 1, 12, 1, 7, 1, 2, 1, 1, 49, 1, 2, 1, 3, 1, 3, 8, 1, 4, 1, 2, 5, 1, 3, 1, 7, 2, 2, 24, 3, 2, 3, 2, 11, 2, 1, 15, 1, 3, 54, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 5, 6
, 2, 7, 2, 1, 1, 8, 1, 1, 5, 1, 2, 2, 4, 3, 8, 2, 1, 1, 8, 43, 1, 1, 1
> , 17, 1, 3, 4, 24, 1, 6, 2, 490, 6, 3, 1, 3, 8, 1, 1, 27, 2, 2, 1, 2, 1, 18, 5, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 14, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 4, 1, 3, 1, 4, 1, 6, 3, 1, 1, 1, 2, 2, 26, 1, 1, 1, 1, 1, 1, 29, 2, 2, 2, 21, 6, 1, 2, 4, 70, 1, 50, 2, 1, 2, 1, 1, 3, 1, 3, 4, 1, 3, 3, 4, 2, 48, 1, 2, 1, 16, 1, 2, 2, 1, 32, 329, 3, 4, 12, 1, 3, 10, 2, 2, 1, 1, 1, 1, 1, 3, 3, 7, 400, 1, 3, 7, 4, 2, 9, 2, 2, 41, 1, 2, 75, 1, 12, 4, 2, 47, 32, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 2, 1, 23, 1, 112, 12, 1, 2, 1, 11, 5, 5, 3, 7, 1, 1, 1, 8, 1, 12, 1, 8, 1, 5, 4, 9, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 4, 1, 2, 2, 1, 2, 38, 1, 2, 1, 2, 2, 38, 59, 1, 8, 4, 2, 1, 4, 2, 3, 1, 29, 19, 2, 369, 1, 1, 7, 1, 5, 9, 2, 3, 1, 1, 3, 2, 2, 34, 1, 1, 1, 11, 1, 3, 1, 8, 1, 10, 1, 1, 1, 32, 1, 3, 1, 1, 8, 4, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 8, 2, 3, 1, 15, 1, 2, 2, 1, 10, 5, 1, 2, 2, 4, 2
9, 8, 5, 1, 1, 2, 5, 1, 2, 1, 2, 6, 1, 1, 1, 11, 2, 5, 1, 2, 115, 6, 14, 4, 121, 1, 4, 2, 15, 34, 24, 6, 1, ...]
>
• ... We should always try to do what most folk think can never be done. Had such an attempt succeeded, there would be several deeply unhappy Fields Medalists,
Message 5 of 9 , Mar 14 12:53 PM
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"WarrenS" <warren.wds@...> wrote:

> I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3
> was a low-degree low-height algebraic number.

We should always try to do what most folk think can never be done.

Had such an attempt succeeded, there would be several deeply
unhappy Fields Medalists, who believe (as I do) in the
Drinfeld-Deligne conjecture that the Grottendieck-Teichmueller
algebra has precisely one generator in each odd degree.

Yet that is, emphatically, not a reason for avoiding serious
experiments to investigate a contrary hypothesis. To the best of
my recollection, the possibility that zeta(3)/Pi^3 might be an
algebraic number has been investigated to many tens of thousands
of decimal digits of numerical precision. However, I am unable
to provide a reference. Might someone else do so, please?

David
• ... --well, you (DJB) probably have better PSLQ and better hardware than I do (I bet) and could do so yourself... I see Broadhurst wrote a quantum field theory
Message 6 of 9 , Mar 15 4:21 PM
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> > Warren D Smith:
> > I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3
> > was a low-degree low-height algebraic number.

> Had such an attempt succeeded, there would be several deeply
> unhappy Fields Medalists, who believe (as I do) in the
> Drinfeld-Deligne conjecture that the Grottendieck-Teichmueller
> algebra has precisely one generator in each odd degree.
>
> Yet that is, emphatically, not a reason for avoiding serious
> experiments to investigate a contrary hypothesis. To the best of
> my recollection, the possibility that zeta(3)/Pi^3 might be an
> algebraic number has been investigated to many tens of thousands
> of decimal digits of numerical precision. However, I am unable
> to provide a reference.

--well, you (DJB) probably have better PSLQ and better hardware
than I do (I bet) and could do so yourself...

I see Broadhurst wrote a quantum field theory paper "where do the tedious products of zetas come from?" (actual title!) which mentions the "Drinfeld Deligne conjecture"
and "GrothendieckTeichm:uller algebra" (whatever they are) as well as using PSLQ.
Unfortunately non-experts (i.e. me) will have a difficult time understanding this paper.
Even after (trying to) read it, I still have almost no clue what DDC and GTA are,
and it gives zero cites to D,D,G and T's work.

Anyhow I agree Zeta(3)/Pi^3 being an "unusual number" was a pretty low-chance proposition, but SM Ruiz's post stimulated me to look, and sure enough, negative result.
About the idea that you can produce miracles on demand if you have enough fitting freedom... well, yes, but the approach I used attempts to quantify the miraculousness,
and the TRUE miracles yield far better approximations than the amount of fitting freedom would suggest. Don't accept lame miracles, demand real miracles.
• ... An editor solicited this article and I agreed on the strict condition that tedious be retained in the title. It happened that I needed to consult it
Message 7 of 9 , Mar 15 4:47 PM
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"WarrenS" <warren.wds@...> wrote:

> Broadhurst wrote a quantum field theory paper
> "where do the tedious products of zetas come from?"

strict condition that "tedious" be retained in the title.
It happened that I needed to consult it yesterday
and fortunately it was easy to google:
> Ungefähr 84 Ergebnisse (0,18 Sekunden)

David
• ... --for example (this is not due to me; perhaps it was first noticed by Ramanujan?) exp(Pi * sqrt(n)) is within 10^(-12) of being an integer if n=163. This
Message 8 of 9 , Mar 15 5:04 PM
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> About the idea that you can produce miracles on demand if you have enough fitting freedom... well, yes, but the approach I used attempts to quantify the miraculousness,
> and the TRUE miracles yield far better approximations than the amount of fitting freedom would suggest. Don't accept lame miracles, demand real miracles.

--for example (this is not due to me; perhaps it was first noticed by Ramanujan?)
exp(Pi * sqrt(n))
is within 10^(-12) of being an integer if n=163.

This is beyond what any kind of fitting contrivance would be expected to produce.
Also, no other n<25000 gets you even within a 100,000 times further away
from being an integer.

PSLQ has in fact discovered truths valid not to 12, but to
an infinite number of decimals :)
• ... Even better: (exp(Pi*sqrt(163))-744)^(1/3) = 640319.999999999999999999999999390... See Cohen, CCANT, p.383, which explains where astonishing results such
Message 9 of 9 , Mar 16 3:31 AM
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--- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> --for example (this is not due to me; perhaps it was first noticed by Ramanujan?)
> exp(Pi * sqrt(n))
> is within 10^(-12) of being an integer if n=163.
>
> This is beyond what any kind of fitting contrivance would be expected to produce.
> Also, no other n<25000 gets you even within a 100,000 times further away
> from being an integer.

Even better:
(exp(Pi*sqrt(163))-744)^(1/3) = 640319.999999999999999999999999390...

See Cohen, CCANT, p.383, which explains where astonishing results such as these come from.

Mike
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