Editing Riesel number (section)

== Riesel problem ==
{{Unsolved|mathematics|Is 509,203 the smallest Riesel number?}}
In 1956, [[Hans Riesel]] showed that there are an [[Infinite set|infinite]] number of integers ''k'' such that <math>k\times2^n-1</math> is not [[prime number|prime]] for any integer&nbsp;''n''. He showed that the number 509203 has this property, as does 509203 plus any positive [[integer]] multiple of 11184810.<ref>{{cite journal | first=Hans | last=Riesel | title=Några stora primtal | journal=Elementa | year=1956 |volume=39 | pages=258–260 | author-link=Hans Riesel}}</ref> The '''Riesel problem''' consists in determining the smallest Riesel number. Because no [[covering set]] has been found for any ''k'' less than 509203, it is [[conjecture]]d to be the smallest Riesel number.

To check if there are ''k'' < 509203, the [[Riesel Sieve|Riesel Sieve project]] (analogous to [[Seventeen or Bust]] for [[Sierpiński number]]s) started with 101 candidates ''k''. As of December 2022, 57 of these ''k'' had been eliminated by Riesel Sieve, [[PrimeGrid]], or outside persons.<ref>{{cite web|url=http://www.primegrid.com/stats_trp_llr.php|title=The Riesel Problem statistics|publisher=PrimeGrid}}</ref> The remaining 42 values of ''k'' that have yielded only composite numbers for all values of ''n'' so far tested are

:23669, 31859, 38473, 46663, 67117, 74699, 81041, 107347, 121889, 129007, 143047, 161669, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 485557, 494743.

The most recent elimination was in April 2023, when 97139 × 2<sup>18397548</sup> −&thinsp;1 was found to be prime by Ryan Propper. This number is 5,538,219 digits long.

As of January 2023, PrimeGrid has searched the remaining candidates up to ''n'' = 14,900,000.<ref>{{Cite web|title=The Riesel Problem statistics|url=https://www.primegrid.com/stats_trp_llr.php|url-status=live|archive-url=https://web.archive.org/web/20240115171321/https://www.primegrid.com/stats_trp_llr.php|archive-date=15 January 2024|access-date=15 January 2024|website=PrimeGrid}}</ref>