Definition:Woodall Prime

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Definition

A Woodall prime is a Woodall number:

$n \times 2^n - 1$

which is also prime.


Sequence

The sequence $\sequence n$ for which $n \times 2^n - 1$ is a prime number begins:

$2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, \ldots$

This sequence is A002234 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The first few of these correspond with the sequence $\sequence n$ of the actual Woodall primes themselves, which begins:

$7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, \ldots$

This sequence is A050918 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also known as

Some sources refer to primes of the form $n \times 2^n - 1$ as Cullen primes, along with those of the form $n \times 2^n + 1$.

However, it is now conventional to reserve the term Cullen primes, named for James Cullen, to those of the form $n \times 2^n + 1$.

The latter are also known as Cunningham primes, for Allan Joseph Champneys Cunningham, so as to ensure their unambiguous distinction from Woodall primes.


Also see


Source of Name

This entry was named for Herbert J. Woodall.


Sources