Math News: Prime Numbers 7

Following the Path of Perfect Numbers

A “Perfect Number” is one whose proper divisors (excluding the number itself) added together equal the number, i.e.:

6: 1,2,3 are its proper divisors.
1+2+3 = 6

Similarly:
28: 1,2,4,7,14 are its proper divisors.
1+2+4+7+14 = 28.

Euclid discovered that Perfect numbers have the form 2n−1(2n − 1).
for n = 2: 21(22 − 1) = 6
for n = 3: 22(23 − 1) = 28
for n = 5: 24(25 − 1) = 496
for n = 7: 26(27 − 1) = 8128.

He also noticed that in each instance above, the expression 2n − 1 is a prime number.
Prime numbers of the form 2n − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers.
As of September 2008, only 46 Perfect Numbers are known associated to the 46 known Mersenne primes.

The first 39 even perfect numbers are 2n−1(2n − 1) for
n = 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 (sequence A000043 in OEIS).
The other 7 known are for n = 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609. It is not known whether there are others between them.

It is still uncertain whether there are infinitely many Mersenne primes and perfect numbers. The search for new Mersenne primes is the goal of the GIMPS distributed computing project.
After the brief explanation of what Perfect Numbers and their related Mersenne Primes are, what follows is an analysis based on the polynomial results being studied here.

Perfect Numbers have the form of the Trinomial final number of terms:
(e2+3e+2)/2 (Jaramillo 1977)

When e takes the values:
Exponent Perfect Number
2 6
6 28
30 496
126 8128
8190 33550336
131070 858969056

Note that the exponent ends in 2, 6 or 0.

Studying the determinants obtained on the sequence of final number of terms of a trinomial raised to a power

has a determinant of 3
has a determinant of 6
has a determinant of 10
has a determinant of 15
has a determinant of 21

Where the sequence is arranged as follows:
Place in Sequence
Trinomial Number of Terms
1 2 3 4 5 6 7
3 6 10 15 21 28 36

The sequence of determinants is equal to the sequence of final number of terms of a trinomial raised to successive integer powers.

If one was to expand the sequence far enough to find the determinants corresponding to the perfect numbers, here is what one would find:

has a determinant of 6 (1st Perfect number)
has a determinant of 28 (2nd Perfect number)
has a determinant of 496 (3rd Perfect number)
has a determinant of 8128 (4th Perfect Number)

Notice that the perfect number itself (determinant) is also found in one of the entries of the matrix.

In general the matrix is:
and its determinant is a Perfect Number P.

Note that the numbers (n+1): 4,8,32,128 are Stirling Numbers of the second kind and the
numbers n: 3,7,21,127 are Mersenne Primes.

Note also that X=P+(n+1). Therefore:
and its determinant is a Perfect Number P.

n N+1

P P+n+1 and its determinant is a Perfect number P.