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Difference between revisions of "Liskovets-Gallot conjectures"
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==Definitions== | ==Definitions== | ||
| − | [[Valery Liskovets]] studied the list of {{Kbn|+|k|n}} primes and observed, that the k's (k divisible by 3) | + | [[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) |
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 <= n <= 100000) | + | Examples: (for 1 <= <var>n</var> <= 100000) |
{| class="wikitable" | {| class="wikitable" | ||
| − | !k-value!!# odd!!# even | + | !<var>k</var>-value!!# odd!!# even |
|- | |- | ||
|51||38||5 | |51||38||5 | ||
| Line 23: | Line 23: | ||
So Liskovets formulated the conjecture: | So Liskovets formulated the conjecture: | ||
| − | :'''"There exist k, 3|k, such that primes {{Kbn|+|k|n}} do exist but only with odd n /only with even n."''' | + | :'''"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>."''' |
| − | [[Yves Gallot]] extended this for {{Kbn|k|n}} numbers and gave also the first solutions as: | + | [[Yves Gallot]] extended this for {{Kbn|<var>k</var>|<var>n</var>}} numbers and gave also the first solutions as: |
| − | :{{Kbn|+|k|n}} is composite for all even | + | :{{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 odd | + | :{{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 even | + | :{{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 odd | + | :{{Kbn|<var>k</var>|<var>n</var>}} is composite for all odd <var>n</var> for <var>k</var>=[[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 k-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 <var>k</var>-values less than the given with the needed condition. |
==Search== | ==Search== | ||
Revision as of 09:26, 5 September 2020
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
- 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
- Problem 36 "The Liskovets-Gallot numbers" from PP&P connection by Carlos Rivera
