Wolfram Language™

Mersenne Primes and Perfect Numbers

A Mersenne prime is a prime number of the form , where the Mersenne prime exponent is itself also a prime number. Each Mersenne prime corresponds to an even perfect number.

Generate a list of Mersenne prime exponents.

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mpe = Table[MersennePrimeExponent[n], {n, 1, 10}]

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Construct the corresponding Mersenne primes.

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mp = 2^mpe - 1

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Construct the corresponding perfect numbers.

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pn = 2^(mpe - 1) (2^mpe - 1)

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AllTrue[pn, PerfectNumberQ]

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Visualize how sparse the distribution of small Mersenne prime exponents is by emphasizing them in red in the list of the first 225 primes.

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primes = Replace[Prime@Range[225], x_?MersennePrimeExponentQ :> Style[x, Red, Bold], 1];

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Multicolumn[primes, Alignment -> {Center, Center}, Spacings -> {1, 1}, Frame -> All, FrameStyle -> Directive[Orange, Dashing[Small]]]

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