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# Liskovets-Gallot 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 exist k, 3|k, such that primes**`k`•2^{n}+1 do exist but only with odd n /only with even n."

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 n=even:**k=66741**`k`•2^{n}+1 is composite for all n=odd:**k=95283**`k`•2^{n}-1 is composite for all n=even:**k=39939**`k`•2^{n}-1 is composite for all n=odd:**k=172677**

## Proof

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

## Search

The current search is maintaind by the Conjectures 'R Us project and can be found here.

## Links

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