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Difference between revisions of "Liskovets-Gallot conjectures"

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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 n=even: '''k=66741'''
+
:{{Kbn|+|k|n}} is composite for all even ''n'' for ''k''=[[Proth prime 66741|66741]]
:{{Kbn|+|k|n}} is composite for all n=odd: '''k=95283'''
+
:{{Kbn|+|k|n}} is composite for all odd ''n'' for ''k''=[[Proth prime 95283|95283]]
:{{Kbn|k|n}} is composite for all n=even: '''k=39939'''
+
:{{Kbn|k|n}} is composite for all even ''n'' for ''k''=[[Riesel prime 39939|39939]]
:{{Kbn|k|n}} is composite for all n=odd: '''k=172677'''
+
:{{Kbn|k|n}} is composite for all odd ''n'' for ''k''=[[Riesel prime 172677|172677]]
  
 
==Proof==
 
==Proof==

Revision as of 19:46, 5 July 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 maintaind by the Conjectures 'R Us project and can be found here.

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