Problems & Puzzles: Conjectures

Conjecture 1. Goldbach's Conjecture 

"In a letter of 1742 to Euler, Goldbach expressed the belief that ‘Every integer N>5 is the sum of three primes’. Euler replied that this is easily seen to be equivalent to the following statement : ‘Every even integer 2n=>4 is the sum of two primes’ (Ref. 1, p. 291) 

Then as we can see the original idea was from Goldbach but the simplification and limitation of it came from Euler. 

By the above reasons the original statement of the Goldbach’s conjecture now is known as "the odd Goldbach conjecture". 

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Samuli Larvala send today (11/08/98) the following interesting information about the status of the work done over this conejcture: 

"Matti Sinisalo has checked the conjecture up to 4*10^11. His paper was published in Math.Comp. "M.K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Math. Comp. 61 (1993)". 

J-M. Deshouillers and Herman te Riele have recently checked it up to 10^14. They published a preview paper on their work when they had reached 10^13. This paper can be found at:
ftp://ftp.cwi.nl/pub/herman/Goldbach/gold13.ps  

On te Riele's web page they say they've now checked it up to 10^14 and the paper will be published soon. The web-page can be found at:
http://dbs.cwi.nl/cwwwi/owa/cwwwi.print_projects?ID=12 (currently broken, 1/9/01) 

Joerg Richstein from the Institute of Informatics, Justus-Liebig-University, announces his results (July 27th, 1998) about "Verifying Goldbach's Conjecture up to 4 x 10¹⁴" at:
http://www.informatik.uni-giessen.de/staff/richstein/ca/Goldbach.html 

M.L. Perez comments (6/7/99) that "The Goldbach Conjecture has been generalized in the form of the  SMARANDACHE CONJECTURE" that can be seen at: http://www.gallup.unm.edu/~smarandache/prim-sum.txt

Jan Felisiak wrote on July 5, 2021:

I would like to say that on 17 May I submitted my article to ScienceOpen under the title "The binary Goldbach conjecture". The paper provides a comprehensive resolution of the Goldbach conjecture as well as the ternary Goldbach conjecture. The ternary conjecture is just a corollary derived from the binary Goldbach conjecture. 

 

The Goldbach comet (puzzle 82) is also discussed in the article and a detailed explanation is provided. The paper also contains a color version of the Goldbach comet, at the very end of the Appendix. The color coding is explained in detail in the paper.

 

The paper also contains some other results which where necessary to prove the Goldbach conjecture. 

 

One more result I think is worth mentioning, at the very end of the paper I proved the Representation of Even Numbers by the Difference of Two Primes. The original conjecture was made by Honwei Shi et al in 2019. The full reference is given in the bibliography.

 

I hope you could post it as a reference for your readership. If you need you may extract some bits for "educational purposes". This I think is necessary for the puzzle 82. 

 

The online address of the article mentioned is:

 

https://www.scienceopen.com/hosted-document?doi=10.14293/S2199-1006.1.SOR-.PPIU46T.v1

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