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.
In[1]:=

mpe = Table[MersennePrimeExponent[n], {n, 1, 10}]
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Construct the corresponding Mersenne primes.
In[2]:=

mp = 2^mpe - 1
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Construct the corresponding perfect numbers.
In[3]:=

pn = 2^(mpe - 1) (2^mpe - 1)
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In[4]:=

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.
In[5]:=

primes = Replace[Prime@Range[225],
x_?MersennePrimeExponentQ :> Style[x, Red, Bold], 1];
In[6]:=

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