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CRUS Liskovets-Gallot
CRUS Liskovets-Gallot is a Conjectures 'R Us (CRUS) subproject aiming to prove the Liskovets-Gallot conjectures, which relate to the smallest Riesel and Proth k-values, divisible by 3, with no primes for n-values of a given parity.
Contents
Explanations
- Main article: Liskovets-Gallot conjectures
Valery Liskovets first observed in 2001 that some k-values, divisible by 3, had few prime n-values of a given parity. He then conjectured that there existed k-values (initially for Proth primes, then also for Riesel primes), divisible by 3, that had no primes of a given parity. This was proven by Yves Gallot, who provided examples for all four cases (Riesel/Proth, even/odd). Gallot further conjectured that these four examples are the smallest such k-values of each type, not including algebraic factorizations.[1]
This subproject is attempting to prove the latter set of conjectures by finding primes for n-values of the required sign (Riesel/Proth) and parity (even/odd). The process is the same as PrimeGrid's subprojects for The Riesel Problem and Seventeen or Bust.
History
This subproject was founded by Conjectures 'R Us in January 2008, as an extension of the base 4 Riesel and Sierpiński problems. The initial search was led by Jean Penné and Gary Barnes, and a page for the effort was created on the Riesel and Proth Prime Database on January 11.
The even Proth conjecture was proven on 2015-08-02, by Penné, and the discovery was made public a day later after a personal double-check.[2]
Current status
Riesel values
Even n's
Odd n's
Proth values
Even n's
- The even n conjecture was proven in August 2015.[2]
Odd n's
- k=9267, reserved by Jean Penné
- k=32247
- k=53133
Primes found
Riesel
Even n's
The data file can be found here.
Odd n's
The data file can be found here.
Proth
Notes
- ↑ Problem 36 "The Liskovets-Gallot numbers" from PP&P connection by Carlos Rivera
- ↑ 2.0 2.1 A Liskovets-Gallot theorem proven! by Jean Penné, 2015-08-02
