New Largest Prime Numbers

Mersenne Primes

Biologically, every male as an infant foetus initially enjoys a feminine identity. What may not be quite so obvious is that this parallels the very nature of the prime number system, which starts with an even (feminine) number.

Indeed, this would suggest that every prime number can ultimately be derived from 2. A famous example of this approach is the set of Mersenne primes. (Another is the set of Fermat primes).

Now Mersenne primes are especially interesting in that they also generate another fascinating class of numbers - with considerable psycho-mathematical significance - i.e. perfect numbers.

A Mersenne prime is always of the form 2n - 1 (n is a positive integer). For example,

25 - 1 = 31 is a Mersenne prime.

Now, there are two points which I wish to point out which illustrate the transrational approach.

1) The power of 2 (i.e. the qualitative vertical number) must itself be prime, if the resulting number (i.e. the reduced quantitative horizontal number) is to be prime. In our example, the qualitative number 5 is prime, and the (reduced) quantitative number 31 is prime.

2) The resulting prime number is closely associated with a highly composite number.

32 (i.e. 25) is the most composite possible for its size. It has five factors all involving the minimum number of 2. Then by reducing by 1, we move to the opposite extreme of a prime number with no factors. Furthermore this seems to be a natural feature of the number system. In other words on either side of the most highly composite numbers (i.e. numbers that are powers of 2), can be found numbers at the opposite extreme with very few factors (frequently being prime). This is just another example of the principle of the complementarity of opposites naturally occuring in the number system. By the addition or subtraction of 1, a highly composite number tends to collapse to its opposite prime or near prime state.

Indeed, we can push this principle to the extreme, by starting from 2, attempting to continually generate prime numbers, through a special recursive process.

Thus 22 - 1 = 3. Therefore, with 2 as the base "qualitative" prime number (i.e. power), through our operation, we generate a new reduced "quantitative" prime.

Now we keep on substituting this derived "quantitative" prime as the new exponent or power. So we keep exchanging "quantitative" for "qualitative" numbers.

Then 23 - 1 = 7, which is the new (reduced) "quantitative" prime. This in turn becomes the new "qualitative" prime, so that,

27 - 1 = 127. This is also prime, and this "quantitative" number, becomes the new "qualitative" number,

with 2127 - 1 = M127. (i.e. 170141183460469231731687303715884105727).

This again is a "quantitative" prime number (with 39 digits).

This automatically leads to the (surely) interesting hypothesis that the next term, which is

2 M127 - 1, (i.e. 2170141183460469231731687303715884105727 - 1) is in turn a prime number (the exponent of 2 with 39 digits), which would be incomparably larger than any yet discovered, ultimately rooted in the 1st root prime "2". Of course we do not have to stop here. We can substitute this "new prime" as the exponent of 2 to generate a further prime

(i.e. 2 M170141183460469231731687303715884105727 - 1) and so on indefinitely.

Furthermore, when we represent the L.H.S in unary form, and the R.H.S. in binary form, the above terms can always be represented by the use of just one digit (i.e 1).

Thus the 1st line can be written

1111 - 1 or 111(unary form) = 11 (binary form) with the second term,

11111111 - 1 or 1111111(unary form) = 111 (binary form) etc.,

Thus what is involved in the above is a continual (ordered) mathematical switching as between binary and unary format generating prime numbers, which can be represented as a system of (differentiated) ones.

As we have seen, in psychological terms, the conscious (horizontal) process is a unary system, while the unconscious (vertical) process is a binary system.

Therefore in complementary manner, the continual (disordered and confused) psychological switching as between binary (unconscious) and unary (conscious) format generates the (purely) instinctive behaviour of the prime structures, which can be represented as a system of undifferentiated oneness.