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Difference between revisions of "Liskovets-Gallot conjectures"

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==Definitions==
 
==Definitions==
−
[[Valery Liskovets]] studied the list of {{Kbn|+|<var>k</var>|<var>n</var>}} primes and observed, that the <var>k</var>'s (<var>k</var> divisible by 3)
+
[[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 <= <var>n</var> <= 100000)
+
Examples: (for 1 ≤ {{Vn}} ≤ 100000)
 
{| class="wikitable"
 
{| class="wikitable"
−
!<var>k</var>-value!!# odd!!# even
+
!{{Vk}}-value!!# odd!!# even
 
|-
 
|-
 
|51||38||5
 
|51||38||5
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So Liskovets formulated the conjecture:
 
So Liskovets formulated the conjecture:
−
<blockquote>'''There exist <var>k</var>, 3|<var>k</var>, such that primes {{Kbn|+|<var>k</var>|<var>n</var>}} do exist but only with odd <var>n</var>/only with even <var>n</var>.'''</blockquote>
+
<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|<var>k</var>|<var>n</var>}} 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|+|<var>k</var>|<var>n</var>}} is composite for all even <var>n</var> for <var>k</var>=[[Proth prime 66741|66741]]
+
:{{Kbn|+|k|n}} is composite for all even {{Vn}} for {{Vk}}=[[Proth prime 66741|66741]]
−
:{{Kbn|+|<var>k</var>|<var>n</var>}} is composite for all odd <var>n</var> for <var>k</var>=[[Proth prime 95283|95283]]
+
:{{Kbn|+|k|n}} is composite for all odd {{Vn}} for {{Vk}}=[[Proth prime 95283|95283]]
−
:{{Kbn|<var>k</var>|<var>n</var>}} is composite for all even <var>n</var> for <var>k</var>=[[Riesel prime 39939|39939]]
+
:{{Kbn|k|n}} is composite for all even {{Vn}} for {{Vk}}=[[Riesel prime 39939|39939]]
−
:{{Kbn|<var>k</var>|<var>n</var>}} is composite for all odd <var>n</var> for <var>k</var>=[[Riesel prime 172677|172677]]
+
:{{Kbn|k|n}} is composite for all odd {{Vn}} for {{Vk}}=[[Riesel prime 172677|172677]]
  
 
==Proof==
 
==Proof==
−
The verification of these conjectures has to be done in the same manner like the [[Riesel problem]]: find a prime for all <var>k</var>-values less than the given with the needed condition.
+
The verification of these conjectures has to be done in the same manner like the [[Riesel problem]]: find a prime for all {{Vk}}-values less than the given with the needed condition.
  
 
==Search==
 
==Search==

Revision as of 16:22, 7 September 2020

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
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 like the Riesel problem: 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 can be found here.

Links

Retrieved from "https://www.rieselprime.de/z/index.php?title=Liskovets-Gallot_conjectures&oldid=11243"