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Liskovets-Gallot 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 n=even: k=66741
- k•2n+1 is composite for all n=odd: k=95283
- k•2n-1 is composite for all n=even: k=39939
- k•2n-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
