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# Liskovets-Gallot conjectures

The **Liskovets-Gallot conjectures** are a family of conjectures regarding the frequency of prime `n`-values of a given parity for Riesel and Proth `k`-values divisible by 3. The notion that certain `k`-values, divisible by 3, have no prime `n`-values of a given parity was first conjectured by Valery Liskovets in 2001. Yves Gallot found suitable non-trivial `k`-values for all four sign/parity combinations shortly thereafter and claimed them to be the smallest such `k`-values, thus forming the conjectures' final form. A search was started by CRUS in 2008 to prove the conjectures by finding primes of the required parity for all smaller `k`-values. The even Proth conjecture was proven in 2015, and CRUS is continuing the CRUS Liskovets-Gallot subproject to find the remaining 9 primes required to prove the other 3 conjectures.

## Contents

## Definitions

Valery Liskovets studied the list of `k`•2^{n}+1 primes and observed, that the `k`'s (`k` divisible by 3)
got an irregular contribution of odd and even exponents yielding a prime.

Examples: (for 1 ≤ `n` ≤ 100000)

k-value |
# odd | # even |
---|---|---|

51 | 38 | 5 |

231 | 51 | 9 |

261 | 56 | 14 |

87 | 2 | 36 |

93 | 1 | 38 |

177 | 8 | 46 |

So Liskovets formulated the conjecture:

There existk, 3|k, such that primesk•2^{n}+1 do exist but only with oddn/only with evenn.

Yves Gallot extended this for `k`•2^{n}-1 numbers and gave also the first solutions as:

`k`•2^{n}+1 is composite for all even`n`for`k`=66741 (proven minimal in 2015)`k`•2^{n}+1 is composite for all odd`n`for`k`=95283`k`•2^{n}-1 is composite for all even`n`for`k`=39939`k`•2^{n}-1 is composite for all odd`n`for`k`=172677

## Proof

The verification of these conjectures has to be done in the same manner as the Riesel and Sierpiński problems: find a prime for all `k`-values less than the given with the needed condition.

## Search

The current search is maintained by the Conjectures 'R Us project and described in the article CRUS Liskovets-Gallot.

## Links

- Problem 36 "The Liskovets-Gallot numbers" from PP&P connection by Carlos Rivera