Gilbreath Conjecture
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
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Preliminary Remark:
The following research is the fruit of a collaboration with Jean-Paul Delahaye professor at the Université de Lille,
researcher at the Lille Cristal laboratory and well known columnist for the Pour La Science newspaper.
1-Definition:
This conjecture was stated in 1958 by orman L. Gilbreath but published earlier in 1878 by François Proth.
It is related to the prime numbers and to the sequences generated by taking the absolute value of
the difference between each prime number and its successor
and then repeating this process ad infinitum:
2 3 5 7 11 13 17 19 23 29 31 (...)
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
1 2 2 4 2 4 2 4 6 2 (...)
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
1 0 2 2 2 2 2 2 4 (...)
\ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ /
1 2 0 0 0 0 0 2 (...)
\ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ /
1 2 0 0 0 0 2 (...)
\ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ /
1 2 0 0 0 2 (...)
\ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ /
1 2 0 0 2 (...)
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
1 2 0 2 (...)
\ / \ / \ / \ /
\ / \ / \ / \ /
\ / \ / \ / \ /
\ / \ / \ / \ /
1 2 2 (...)
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
1 0 (...)
\ / \ /
\ / \ /
\ / \ /
\ / \ /
1 (...)
\ /
\ /
\ /
\ /
(...)
The conjecture states that the first value of each line is 1 (except the first one where it is a 2 -the only even prime number-)
and was studied by Andrew Odlyzko in 1993. He did check it for all prime numbers less than 1013.
On sunday 10/05/2025 20:45 (Paris time, France) I did succeed to check it up to 1014.
2-IN PROGRESS/EN COURS
IN PROGRESS/EN COURS
- [01]
IN PROGRESS/EN COURS
- [02]
IN PROGRESS/EN COURS
Copyright © Jean-François COLONNA, 2025-2025.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2025-2025.
