Riesel Problem

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Unsolved Problem

What is the smallest Riesel number?


Source of Name

This entry was named for Hans Ivar Riesel.


Historical Note

As of 13 December 2017, there remain $49$ values of $k$ below $509 \, 203$ for which no primes have been found of the form:

$k 2^n - 1$

for any positive integer $n$.


They are as follows:

$2293, 9221, 23669, 31859, 38473,$
$46663, 67117, 74699, 81041, 93839,$
$97139, 107347, 121889, 129007, 143047,$
$146561, 161669, 192971, 206039, 206231,$
$215443, 226153, 234343, 245561, 250027,$
$315929, 319511, 324011, 325123,$
$327671, 336839, 342847, 344759, 362609,$
$363343, 364903, 365159, 368411, 371893,$
$384539, 386801, 397027, 409753, 444637,$
$470173, 474491, 477583, 485557, 494743$

Any of these may therefore be a Riesel number smaller than $509 \, 203$.

Research is ongoing.


Sources