IsMersennePrimeExponent

Tests whether 2^p - 1 is prime, by the Lucas-Lehmer test.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMersennePrimeExponentQ✓

IsMersennePrimeExponent(p)True if p is prime and 2^p - 1 passes Lucas-Lehmer, False if p is not prime or fails it

Typed signatures

Type	Package	Applies
`(integer) -> boolean`	`combinatorics`	always

Domain: Collections

Details

Runs Lucas-Lehmer directly (s0 = 4, s(i+1) = s(i)^2 - 2 mod 2^p - 1, prime iff s(p-2) = 0) rather than a table lookup -- exact, not a probable-prime test.
Capped at p <= 4423 (the 20th known Mersenne prime exponent, same bound number-theory's own MersennePrimeExponent table is complete to) -- past that, whether 2^p - 1 is prime is GIMPS-scale search, so the call stays unevaluated rather than running an arbitrarily expensive test.
A non-prime p (or p < 2) answers False outright, without running Lucas-Lehmer -- a composite exponent's Mersenne number always has a proper factor.

Examples

Basic #

In

\mathrm{IsMersennePrimeExponent}(7)

Out

\mathrm{True}

$2^5 - 1 = 31$ is prime

In

\mathrm{IsMersennePrimeExponent}(5)

Out

\mathrm{True}

A composite exponent never works: $2^8 - 1 = 255 = 3 \cdot 5 \cdot 17$

In

\mathrm{IsMersennePrimeExponent}(8)

Out

\mathrm{False}

Scope #

In

\mathrm{IsMersennePrimeExponent}(11)

Out

\mathrm{False}

In

\mathrm{IsMersennePrimeExponent}(9)

Out

\mathrm{False}

In

\mathrm{IsMersennePrimeExponent}(31)

Out

\mathrm{True}

Negative integers are not exponents of Mersenne primes

In

\mathrm{IsMersennePrimeExponent}(-5)

Out

\mathrm{False}

Applications #

$2^{89} - 1$ is a Mersenne prime, a large one

In

\mathrm{IsMersennePrimeExponent}(89)

Out

\mathrm{True}

Properties #

Even exponents above $2$ fail since $2^{2k} - 1$ factors

In

\mathrm{IsMersennePrimeExponent}(12)

Out

\mathrm{False}

The only even exponent: $2^2 - 1 = 3$

In

\mathrm{IsMersennePrimeExponent}(2)

Out

\mathrm{True}

Possible issues #

With an unknown argument the test stays unevaluated

In

\mathrm{IsMersennePrimeExponent}(\mathrm{FullSimplify}(x))

Out

\mathrm{IsMersennePrimeExponent}(x)

Neat examples #

$2^{13} - 1 = 8191$ , a prime

In

\mathrm{IsMersennePrimeExponent}(13)

Out

\mathrm{True}

About

Examples	12
Checked against	Wolfram Language 12 agree
Crosswalk	Wolfram Language

See also: MersennePrimeExponent, IsPrime