Sierpiński Problem

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Unsolved Problem

What is the smallest Sierpiński number of the second kind?


Historical Note

It was proved by John Lewis Selfridge in $1962$ that $78\ 557$ is a Sierpiński number of the second kind.

In 1967, Wacław Franciszek Sierpiński and John Selfridge conjectured that $78\ 557$ is in fact the smallest Sierpiński number of the second kind.


To prove that this is the case, all odd positive integers smaller than $78\ 557$ are not Sierpiński numbers of the second kind.

That is, it must be shown that, for all $1 \le k \le 78\ 557$, where $k$ is odd, there exists some $n \in \Z$ such that:

$k 2^n + 1$

is prime.


By March 2002, there were seventeen such $k$ whose status was still unknown.

That was when the distributed computing project Seventeen or Bust was established.

Its aim was to test all these remaining seventeen numbers by exhaustively checking all values of $n$ until finding a value of $n$ for which $k 2^n + 1$ is prime.

In April 2016 the project terminated as a result of the server going down, with the resulting loss of both the server and the backups. Work continues on PrimeGrid.


As of 18 January 2017, twelve of those remaining seventeen numbers have been found to be non-Sierpiński, by establishing a value of $n$ for which $k 2^n + 1$ is prime, as follows:

$k$ $n$ Number of digits in $k 2^n + 1$ Date discovered Discovered by
$\ 4847\ $ $\ 3\ 321\ 063\ $ $\ 999\ 744\ $ 15 October 2005 Richard Hassler
$\ 5359\ $ $\ 5\ 054\ 502\ $ $\ 1\ 521\ 561\ $ 6 December 2003 Randy Sundquist
$\ 10\ 223\ $ $\ 31\ 172\ 165\ $ $\ 9\ 383\ 761\ $ 31 October 2016 Szabolcs Peter
$\ 19\ 249\ $ $\ 13\ 018\ 586\ $ $\ 3\ 918\ 990\ $ 5 May 2007 Konstantin Agafonov
$\ 21\ 181\ $ $\ > 17\ 000\ 011\ $ Search ongoing
$\ 22\ 699\ $ $\ > 17\ 008\ 677\ $ Search ongoing
$\ 24\ 737\ $ $\ > 17\ 001\ 366\ $ Search ongoing
$\ 27\ 653\ $ $\ 9\ 167\ 433\ $ $\ 2\ 759\ 677\ $ 8 June 2005 Derek Gordon
$\ 28\ 433\ $ $\ 7\ 830\ 457\ $ $\ 2\ 357\ 207\ $ 30 December 2004 anonymous
$\ 33\ 661\ $ $\ 7\ 031\ 232\ $ $\ 2\ 116\ 617\ $ 17 October 2007 Sturle Sunde
$\ 44\ 131\ $ $\ 995\ 972\ $ $\ 299\ 823\ $ 6 December 2002 anonymous
$\ 46\ 157\ $ $\ 698\ 207\ $ $\ 210\ 186\ $ 27 November 2002 Stephen Gibson
$\ 54\ 767\ $ $\ 1\ 337\ 287\ $ $\ 402\ 569\ $ 22 December 2002 Peter Coels
$\ 55\ 459\ $ $\ > 17\ 000\ 193\ $ Search ongoing
$\ 65\ 567\ $ $\ 1\ 013\ 803\ $ $\ 305\ 190\ $ 3 December 2002 James Burt
$\ 67\ 607\ $ $\ > 17\ 000\ 090\ $ Search ongoing
$\ 69\ 109\ $ $\ 1\ 157\ 446\ $ $\ 348\ 431\ $ 7 December 2002 Sean DiMichele


Source of Name

This entry was named for Wacław Franciszek Sierpiński.


Sources

The Sierpinski Problem: Definition and Status