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Difference between revisions of "Liskovets-Gallot conjectures"
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| + | The '''Liskovets-Gallot conjectures''' are a family of conjectures regarding the frequency of prime {{Vn}}-values of a given parity for Riesel and Proth {{Vk}}-values divisible by 3. The notion that certain {{Vk}}-values, divisible by 3, have no prime {{Vn}}-values of a given parity was first conjectured by [[Valery Liskovets]] in 2001. [[Yves Gallot]] found suitable non-trivial {{Vk}}-values for all four sign/parity combinations shortly thereafter and claimed them to be the smallest such {{Vk}}-values, thus forming the conjectures' final form. A search was started by [[Conjectures 'R Us|CRUS]] in 2008 to prove the conjectures by finding primes of the required parity for all smaller {{Vk}}-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. | ||
| + | |||
==Definitions== | ==Definitions== | ||
| − | [[Valery Liskovets]] studied the list of {{Kbn|+|k|n}} primes and observed, that the | + | [[Valery Liskovets]] studied the list of {{Kbn|+|k|n}} primes and observed, that the {{Vk}}'s ({{Vk}} divisible by 3) |
got an irregular contribution of odd and even exponents yielding a prime. | got an irregular contribution of odd and even exponents yielding a prime. | ||
| − | Examples: (for 1 | + | Examples: (for 1 ≤ {{Vn}} ≤ 100000) |
{| class="wikitable" | {| class="wikitable" | ||
| − | ! | + | !{{Vk}}-value!!# odd!!# even |
|- | |- | ||
|51||38||5 | |51||38||5 | ||
| Line 23: | Line 25: | ||
So Liskovets formulated the conjecture: | So Liskovets formulated the conjecture: | ||
| − | + | <blockquote>'''There exist {{Vk}}, 3|{{Vk}}, such that primes {{Kbn|+|k|n}} do exist but only with odd {{Vn}}/only with even {{Vn}}.'''</blockquote> | |
[[Yves Gallot]] extended this for {{Kbn|k|n}} numbers and gave also the first solutions as: | [[Yves Gallot]] extended this for {{Kbn|k|n}} numbers and gave also the first solutions as: | ||
| − | :{{Kbn|+|k|n}} is composite for all | + | :{{Kbn|+|k|n}} is composite for all even {{Vn}} for {{Vk}}=[[Proth prime 2 66741|66741]] (proven minimal in 2015) |
| − | :{{Kbn|+|k|n}} is composite for all | + | :{{Kbn|+|k|n}} is composite for all odd {{Vn}} for {{Vk}}=[[Proth prime 2 95283|95283]] |
| − | :{{Kbn|k|n}} is composite for all | + | :{{Kbn|k|n}} is composite for all even {{Vn}} for {{Vk}}=[[Riesel prime 2 39939|39939]] |
| − | :{{Kbn|k|n}} is composite for all | + | :{{Kbn|k|n}} is composite for all odd {{Vn}} for {{Vk}}=[[Riesel prime 2 172677|172677]] |
==Proof== | ==Proof== | ||
| − | The verification of these conjectures has to be done in the same manner | + | The verification of these conjectures has to be done in the same manner as the [[Riesel problem 1|Riesel]] and [[Sierpiński problem]]s: find a prime for all {{Vk}}-values less than the given with the needed condition. |
==Search== | ==Search== | ||
| − | The current search is | + | The current search is maintained by the [[Conjectures 'R Us]] project and described in the article [[CRUS Liskovets-Gallot]]. |
==Links== | ==Links== | ||
*[https://www.primepuzzles.net/problems/prob_036.htm Problem 36] "The Liskovets-Gallot numbers" from [https://www.primepuzzles.net/index.shtml PP&P connection] by [[Carlos Rivera]] | *[https://www.primepuzzles.net/problems/prob_036.htm Problem 36] "The Liskovets-Gallot numbers" from [https://www.primepuzzles.net/index.shtml PP&P connection] by [[Carlos Rivera]] | ||
| − | [[Category: | + | [[Category:Conjectures]] |
Latest revision as of 12:42, 9 May 2024
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•2n+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•2n+1 do exist but only with odd n/only with even n.
Yves Gallot extended this for k•2n-1 numbers and gave also the first solutions as:
- k•2n+1 is composite for all even n for k=66741 (proven minimal in 2015)
- k•2n+1 is composite for all odd n for k=95283
- k•2n-1 is composite for all even n for k=39939
- k•2n-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
