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}]

Out[1]=

Construct the corresponding Mersenne primes.

In[2]:=

mp = 2^mpe - 1

Out[2]=

Construct the corresponding perfect numbers.

In[3]:=

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

Out[3]=

In[4]:=

AllTrue[pn, PerfectNumberQ]

Out[4]=

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]]]

Out[6]=