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]:=
![Click for copyable input](assets.en/mersenne-primes-and-perfect-numbers/In_1.png)
mpe = Table[MersennePrimeExponent[n], {n, 1, 10}]
Out[1]=
![](assets.en/mersenne-primes-and-perfect-numbers/O_1.png)
Construct the corresponding Mersenne primes.
In[2]:=
![Click for copyable input](assets.en/mersenne-primes-and-perfect-numbers/In_2.png)
mp = 2^mpe - 1
Out[2]=
![](assets.en/mersenne-primes-and-perfect-numbers/O_2.png)
Construct the corresponding perfect numbers.
In[3]:=
![Click for copyable input](assets.en/mersenne-primes-and-perfect-numbers/In_3.png)
pn = 2^(mpe - 1) (2^mpe - 1)
Out[3]=
![](assets.en/mersenne-primes-and-perfect-numbers/O_3.png)
In[4]:=
![Click for copyable input](assets.en/mersenne-primes-and-perfect-numbers/In_4.png)
AllTrue[pn, PerfectNumberQ]
Out[4]=
![](assets.en/mersenne-primes-and-perfect-numbers/O_4.png)
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]:=
![Click for copyable input](assets.en/mersenne-primes-and-perfect-numbers/In_5.png)
primes = Replace[Prime@Range[225],
x_?MersennePrimeExponentQ :> Style[x, Red, Bold], 1];
In[6]:=
![Click for copyable input](assets.en/mersenne-primes-and-perfect-numbers/In_6.png)
Multicolumn[primes, Alignment -> {Center, Center}, Spacings -> {1, 1},
Frame -> All, FrameStyle -> Directive[Orange, Dashing[Small]]]
Out[6]=
![](assets.en/mersenne-primes-and-perfect-numbers/O_5.png)