Conjecture: “If the successive
(increasing & odd)
primes are subtracted from an even number, and the number of trials needed
to obtain another prime is divided for the square of the natural logarithm
of the even number , this relation taken from the case of large number of
primes subtracted, it is bounded by a constant
as the even numbers grows
to infinite”
Rodríguez argues that 'this
conjecture, naturally, reinforces the Goldbach’s conjecture'
Example: Be 1077422 an even number. If
we subtract the primes 3,5,7,11...599 from that number, the remainders are
composite numbers. Only when the subtrahend attains 601 then the remainder
results a prime = 1076811.
The number of trials is precisely the
rank of 601 in the sequence of primes, that is 109. Relation = 109 /
(log(1077422))^2 = 0.565. We will call this relation Quality Index.
The plausibility of the conjecture
written above is based in the following data, extracted from the works of
Sinisalo, Deshouillers, Te Riele, Saouter,
Richstein and T. Oliveira (*)
Rodríguez provides the following Table
that summarizes the empirical search done around this issue.
LARGE NUMBER OF TRIALS NEEDED TO
PRODUCE A PRIME DIFFERENCE WHEN THE SUCCESSIVE PRIMES ARE
SUBTRACTED FROM EVEN NUMBERS
(Only the indexes
larger than 0.62 has been considered)
Even Number
Trials
Quality
Date
Author
419911924
249 0.6316 1993 SINISALO
721013438
277 0.6659 1993 SINISALO
1847133842
283 0.6216 1993 SINISALO
35884080836
408 0.6907 1993 SINISALO
599533546358
482
0.6554
1998 DESHOUILLERS
76903574497118
655 0.6407 “
“
184162477860248
677 0.6275 “
“
217361316706568
692 0.6305 “
“
389965026819938
734 0.6503 “
“
1047610575836828
838
0.7006 2001 T. OLIVEIRA
24925556008175266 958
0.6729
"
"
31284177910528922
982
0.6807
2004
"
43181037765133228 958
0.6529 “ “
Questions:
1. Do you devise an
argument on favor of the given conjecture?
If so, what would be the value for the constant supposed to tend the
quotient of this conjecture?
2. Can you get
the next even number
needing more than 982
trials?
3. Can you get
the next even number
with an index larger than 0.7006?
__________
(*)
SINISALO M.K,
‘Checking the Goldbach’s Conjecture up to
4 x
10^11,’Mathematics
of Computation Vol. 61 - 1993.
DESHOUILLERS , te RIELE,
SOUTER ‘New Experimental Results Concerning the
Goldbach Conjecture’ - Algorithmic Number Theory,
Third International Symposium – Portland ORE,
1998.
RICHSTEIN
.G, 'Goldbach’s
Conjecture to 4x10^14’,
Mathematics of
Computation Vol 70 Oct. 2001.
T. OLIVEIRA.
Enoch Haga and
Farideh Firoozbakht helped to correct some minor typos in the original
Table sent by Rodríguez. I have incorporated these corrections to
the Table above in order not to handle two Tables.
***
Here is a contribution from
Didier van der Straten from Belgium
(April, 2005):
I am just an amateur, interested in every
recent finding about Goldbach conjecture.
I made several math researches, using small
figures, after what I suspected a good direction was to investigate
"gaps between Goldbach partitions".
Then I fell on your text Conjecture 36 by
Sinisalo-Ludovicus, and the questions which seem to remain outstanding.
I soon realised that my study would need to
go to that sort of order of magnitude to realise any progress.
I am working at that. So any data about
those gaps is of interest for me. That includes the initial gap before
reaching a minimal partition.
Meanwhile I kept searching about Goldbach
and I found a page which is in line with sbj conjecture annd its
questions
Question 2 : Answer is yes more than 982
trials has been found. See that tos page.
Question 3 (the most interesting) : No but
in that range, numbers with higher trial values seem to stay oscillating
very close to that maximum, without reaching
it.
It would be nice that you update your page
with that conntribution of Tomas.
In my opinion I am convinced this is a good
direction.
I hope this will help the Golbach fans.
Please also contact Sinisalo and/or Ludovicus and/or Tomas.
***
Luis Rodríguez wrote (16/04/05):
Siguiendo el consejo de Didier,
busqué en la dirección de Oliveira y efectivamente hay dos valores que
vale la pena añadir a la tabla. Así:
EVEN NUMBER = 121 005 022 304 007
026
TRIALS .... = 1056
QUALITY ... = 0.6825
DATE ...... = 2004
AUTHOR .... = SIEGFRIED HERTZOG
EVEN NUMBER = 258 549 426 916 149 682
TRIALS .... = 1111
QUALITY ... = 0.6911
DATE ...... = 2005
AUTHOR .... = SIEGFRIED HERTZOG
***
On Dec 08, Luis Rodríguez added:
From Tomas Oliveira : http://ieeta.pt/~tos/goldbach.html
we have this new value.
Even number = 906,030,579,562,279,642
Rank of prime subtracted = 1156
Quality Index = 0.6761
Authors = Fettig & Sobh
All even numbers less than 4x10^18 has been
tested. Date 12-25-2008
***
