Problems & Puzzles: Conjectures

Conjecture 36. The Sinisalo-Ludovicus Conjecture

Luis Rodríguez has sent the following conjecture:

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                “                     “
21736131
6706568      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.
.

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.

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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

http://www.ieeta.pt/~tos/goldbach.html by Tomas Oliveira e Silva.

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.

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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

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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

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