The Prime Glossary: Colbert number

Colbert number
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The set of the Colbert Report

A Colbert number is any megaprime whose discovery contributes to the long sought-after proof that k = 78557 is the smallest Sierpinski number. These are whimsically named after Stephen T. Colbert, the American comedian, satirist, actor and writer. There are currently only five known Colbert Numbers:

19249^.2^13018586+1	(**)	3,918,990 digits	(#)
27653^.2^9167433+1	(**)	2,759,677 digits	(#)
28433^.2^7830457+1	(**)	2,357,207 digits	(#)
33661^.2^7031232+1	(**)	2,116,617 digits	(#)
5359^.2^5054502+1	(**)	1,521,561 digits	(#)

Waclaw Sierpinski used modular covers to prove there were infinitely many odd integers k for which k^.2ⁿ+1 are composite for all n > 1. It has been conjectured that 78557 is the smallest, but to prove that requires a truly massive amount of computation--we must find primes of the form k2^.ⁿ+1 for all small k values. Most of those primes were easily found, but the final six (below), have so far defied all attempts.

This search has been going on for over four decades! The primes above were found by the dramatically successful project Seventeen or Bust. Their name came from the fact that there were 17 primes left to find when they began. Due to their work, there are now just six:

10223^.2^???+1, 21181^.2^???+1, 22699^.2^???+1, 24737^.2^???+1, 55459^.2^???+1, 67607^.2^???+1

When such primes are found (and we expect they will be), these will all be Colbert numbers. It is likely that the largest of these will be larger than the largest currently known prime.

See Also: SierpinskiNumber

Related pages (outside of this work)

Seventeen or Bust --A Distributed Attack on the Sierpinski Problem
The Colbert Nation --The Colbert Report Official Site

References:

HMSW2008
L. Helm, P. Moore, P. Samidoost and G. Woltman, "Resolution of the mixed Sierpinski problem," Integers: Elect. J. Comb. Num. Theory, 8:A61 (2008)
Sierpinski60
W. Sierpinski, "Sur un probléme concernment les nombres k · 2ⁿ +1," Elem. Math., 15 (1960) 63-74. [See the glossary entry Sierpinski number as well as the paper [Riesel56].]