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Difference between revisions of "Liskovets-Gallot conjectures"

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(link)
(mention which case is already proven; mention Sierpiński problem as well; avoid "here" link)
 
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[[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 even {{Vn}} for {{Vk}}=[[Proth prime 2 66741|66741]]
+
:{{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 odd {{Vn}} for {{Vk}}=[[Proth prime 2 95283|95283]]
 
:{{Kbn|+|k|n}} is composite for all odd {{Vn}} for {{Vk}}=[[Proth prime 2 95283|95283]]
 
:{{Kbn|k|n}} is composite for all even {{Vn}} for {{Vk}}=[[Riesel prime 2 39939|39939]]
 
:{{Kbn|k|n}} is composite for all even {{Vn}} for {{Vk}}=[[Riesel prime 2 39939|39939]]
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==Proof==
 
==Proof==
The verification of these conjectures has to be done in the same manner like the [[Riesel problem 1|Riesel problem]]: find a prime for all {{Vk}}-values less than the given with the needed condition.
+
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 maintained by the [[Conjectures 'R Us]] project and can be found [[CRUS Liskovets-Gallot|here]].
+
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:Conjectures]]
 
[[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.

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