49 prime numbers

Not all prime numbers are created equal.  Some are more "perfect" than the others.  What do the following 49 prime numbers have in common?

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281

Each gives rise to one of the 49 known Mersenne primes Mp when substituted for p in the calculation

.

Mersenne primes are closely related to perfect numbers.

Prime generating formulas?

I wrote about Wilson's theorem previously.  While Wilson's theorem can be used to test whether a positive integer is prime, there are no known formulas generating prime numbers.  Throughout the ages, mathematicians have been searching for such a formula, yet without success.  In addition to the Mersenne primes, there are other numbers studied by Fermat and Euler.

The Fermat numbers are generated by the formula, for n = 0, 1, 2, ...

The first five Fermat numbers 3, 5, 17, 257 and 65537 are all prime.  However, when n = 5,

F₅ = 2³² + 1 = 4294967297 = 641 × 6700417

is composite.

Euler saw that the formula n² − n + 41 generates primes for n = 1, 2, 3, ... , 40.  Clearly, the formula fails to generate a prime when n = 41.

M = 24N + 7

Summer is over.  Can't believe I haven't posted anything on math for three months.  So here it is ...

Further to my previous post on Mersenne primes, here is another neat fact about these special prime numbers.  All Mersenne primes M, except for the first one, are congruent to 7 modulo 24.  In other words, they can be written in the form M = 24N + 7 for some integer N.  For example,

M3 = 7 = 24 x 0 + 7
M5 = 31 = 24 x 1 + 7
M7 = 127 = 24 x 5 + 7
M13 = 8191 = 24 x 341 + 7

Perfect numbers

A number is said to be perfect if it is equal to the sum of all its divisors, excluding itself.  Perfect numbers have been studied by ancient Greek mathematicians including Euclid.  They are closely related to Mersenne primes.  Euler has proved that all even perfect numbers can be written as:

for all prime p associated with a Mersenne prime.  The last expression above hints that even perfect numbers must be triangular.  The first few even perfect numbers, when p = 2, 3, 5, 7, 13, are:

6, 28, 496, 8128, 33550336.

It is not known whether odd perfect numbers exist.

Mersenne primes

The search for prime numbers has been a constant endeavour for curious minds since ancient times.  For centuries, mathematicians have been trying, in vain, to find a formula that generates prime numbers.  Mersenne primes are prime numbers of the form

where p is prime.  Not all Mersenne numbers are prime.  For instance, primes are generated when p = 2, 3, 5, 7, 11, 13:

3, 7, 31, 127, 2047, 8191

2047 = 23 x 89 is not prime.