On random numbers
Introduction
I have had this worked out for about 5 years now, along with the Riemann theory. So I thought I would sketch out the general ideas here and follow up with a more formal proof later.
In my paper on the Riemann Hypothesis I showed (I think) that the integers can be regarded as random numbers, because if you were selecting numbers randomly you could select the integers in ascending order. Not all number patterns can be selected randomly. For example if you select N numbers randomly half might be even, and the more you select the more likely half are even. For any finite number of selections half might not be even, but for all numbers, as N goes to infinity half must be even or the sample not random.
By this I mean if you continue to select numbers and never stop then half of them will be even or the numbers were not selected randomly.
So the integers (in ascending order if not otherwise specified) have half of them even, 1/3 have 3 as a factor, and generally 1/X of them have X as a factor. For a random selection of N numbers as N goes to infinity, also 1/X of them have X as a factor.
So if one is selecting N numbers randomly and keeps going one should be able to select the integers. Also not every set of N numbers selected as N goes to infinity is random.
So the set of random number sequences or RNS contains all sequences that are infinite and which can be selected randomly. The set of nonrandom number sequences or NRNS cannot be selected randomly. The integers are a sequence in RNS.
So the integers should have certain properties being in the set of random number selections. Nonrandom number selections should have some different properties. This paper then begins to explore these aspects with prime numbers. To do this generally only odd numbers will be selected.
(1) Generally then as one selects random numbers 1/X of them tend to have X as a factor.
With the integers this fraction is deviated from least of all, that is it alternates between even and odd. So it is not possible for a sequence in the set of random numbers to deviate less from (1) than the integers do. But this does not exclude the integers from that set.
Generally then we can perform any test on the integers by doing the same test on a sequence from this set of random number sequences. While some results may deviate in the short term the results will be the same for the complete sequence.
Not all the prime patterns below are explained according to this theory, but it should apply to them all. I will be adding more prime patterns in the next version of this paper.
twin, sexy and cousin primes
This can be viewed first by discarding all even numbers selected randomly, so that all remaining are odd. Now it seems that when a prime number is selected if the selection is random then the next number selected might also be prime. For any finite number of selections the next number selected might not be prime, but at each prime selection there is a fraction of numbers remaining prime/composite. So the odds are that on the next selection a prime selection is possible.
So is it then possible to have a random selection of odd numbers so that no two in a row are prime? It appears no. For example if a prime is selected, then there is a fraction of remaining numbers that is prime, and if the next number selected cannot be prime then the numbers selected are not random. Of course the next number might not be prime, and there might be a run of composite numbers of any finite length.
So for the integers, if after a number N there are no prime pairs then the integers are not random which is false, therefore there must be another pair of twin primes. The same applies for cousin and sexy primes or generally any prime pattern not sieved out as impossible. A pattern like 3,5,7 occurs once as 3 primes each separated by two. Randomly there can also be 3 primes selected in a row but the chance of this occurring goes to zero as N goes to infinity. Since the odds of two primes in a row cannot go to zero there must be an infinite number of these prime patterns.
Quotes are taken from Prime numbers, the most mysterious figures in maths by D. Wells.
arithmetic progressions, of primes
In an arithmetic progression (or sequence) the differences between
successive terms are constant, for example:
3 7 11 15 19 23 27 31 35 39 43 ...
with constant difference 4. This happens to already contain seven
primes, with one sequence of three consecutive primes.
Again this same method could be done with a random number sequence. So for example one might discard every second number selected, and again the Nth term selected randomly will tend the have the same factors as the Nth integer. So again a prime would be selected randomly infinitely often in the progression.
Cullen primes
Numbers of the form Cn = n _ 2^n + 1 are named after the Reverend J.
Cullen, who noticed in 1905 that apart from C1 = 3 and one other
possible exception, they are all composite for n = 1 to 100. The
exception was C53, which was found by Cunningham to be divisible
by 5591.
Although for low values of n, Cullen primes are rare, it has been
conjectured that there is an infinite number of them.
Again these should contain an infinite number of primes from a random sequence.
Dickson’s conjecture
Leonard Eugene Dickson (1874–1954) is best known today for his
extraordinarily detailed three-volume History of the Theory of Numbers,
whose first volume is on Divisibility and Primality. He conjectured
in 1904 that if we have a sequence of linear expressions with
integer coefficients, with all the ai positive,
a1n + b1, a2n + b2, a3n + b3, . . . , akn + bk
then there is infinite number of values of n for which every one of these
expressions will be prime simultaneously (apart from cases where
there is a number which divides at least one the expressions for every
value of n).
Generally this should be true with random numbers so should be true with the integers.
Dirichlet’s theorem and primes
in arithmetic series
Euclid proved that there is an infinity of primes among the positive
integers, but how many are there in arithmetic progressions such as
44 • Dirichlet’s theorem and primes in arithmetic series
1 5 9 13 17 21 25 29 . . .
or 2 7 12 17 22 27 32 37 . . . ?
Dirichlet proved in 1837, a conjecture made by Gauss: if a and b are
coprime positive integers, then the arithmetic progression a, a + b,
a + 2b, a + 3b, . . . contains infinitely many primes.
primes in polynomials
Dirichlet also proved that if a, 2b, and c have no common prime factor,
then the quadratic expression ax2 + 2bxy + cy2 takes an infinity
of prime values.
See Hardy; Hardy-Littlewood conjectures
primes of the form 4n + 3
We can use Euclid’s method slightly adapted to prove that there is an
infinity of primes of the form 4n + 3.
The product of several numbers of the form 4n + 1 is always of the
same form, and the product of an odd number of numbers of the
form 4n + 3 is also of the form 4n + 3: but the product of an even
number of numbers of the form 4n + 3 is of the form 4n + 1.
Therefore, if we multiply, say, the first six terms in the 4n + 3
sequence,
3 7 11 19 23 31 43 47 ...
and add 2: 3 _ 7 _ 11 _ 19 _ 23 _ 31 + 2
we get a number of the form 4n + 3 that is not divisible by 3, 7, 11,
19, 23, or 31. Its prime factors cannot all be of the form 4n + 1, so at
least one factor must be a “new” prime of the form 4n + 3. Call it p,
and form the expression,
32 _ 7 _ 11 _ 19 _ 23 _ 31 _ p + 2
Once again we have a product of an even number of numbers, 4n +
3, with a factor of that form. So there is an infinity of 4n + 3 primes.
Matching methods show that there is an infinity of primes of the
form 5n + 4, 8n + 3, 8n + 5, and 8n + 7. (Sierpinski)
factorial primes
Factorial primes are of the form n! _ 1. Both forms have been tested
to n = 10,000:
• n! + 1 is prime for n = 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154,
320, 340, 399, 427, 872, 1477, 6380, 26951, . . .
• n! − 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,
324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480,
34790, . . .
Fermat’s conjecture, Fermat numbers,
and Fermat primes
Fermat also studied the sequence of numbers 22n + 1. They are now
known as Fermat numbers. The first few are:
n 0 1 2 3 4 5
Fn 3 5 17 257 65537 composite
Fermat knew that the first five are all prime, and he famously conjectured
that they all are: which turned out to be an equally famous
mistake.
Fibonacci numbers
It isn’t known if there are an infinite number of Fibonacci primes.
This should also extend to sequence such as the Fibonacci and Lucas numbers. So if a set of random numbers is selected one might select the 1,3,5,8,13,… number from this set and one should be able to select a prime number infinitely often. So with any sequence in the set of random number sequences one can select the 3rd, 5th, 8th, … and generally the factors of the Nth Fibonacci number will tend toward the same as the Nth number selected this way. Since again there is no probability stopping primes in this sequence there must be an infinite number of primes.
Generalized Fermat numbers
These are numbers of the form, a2n + b2n. The Generalized Fermat
Prime Search is organized by Phil Carmody using a new and powerful
test, which enables Generalized Fermat primes to be found as
quickly as Mersenne primes of the same size. If you fancy joining in,
here are some of the records you have to beat!
k-tuples conjecture, prime
How many triples of consecutive primes are there? We exclude 3-5-7
because it is one of a kind. Also, the remainders when we divide the
pattern n, n + 2, and n + 4 by 3 must be 0, 2, and 1 in that cyclic
order, so one of them is always divisible by 3.
So we focus on triples such as 5-7-11 and 7-11-13, of the form n,
n + 2 or n + 4, n + 6. They exist for these values of n:
5, 7, 11, 13, 17, 37, 41, 67, 97, 101, 103, 107, 191, 193, 223, 227,
277, 307, 347, 457, 613, 641, . . . (Sloane A007529)
Of these, ten are of the n, n + 2, n + 6 form and twelve are of the n,
n + 4, n + 6 form. Up to 108 there are 55,600 triples of the form (p,
p + 2, p + 6) and 55,556 of the form (p, p + 4, p + 6). (Caldwell, Prime
Pages) The Hardy-Littlewood prime
Legendre, A. M. (1752–1833)
Legendre discussed the law of quadratic reciprocity, and mistakenly
believed that he had proved a theorem that was later proved by
Dirichlet: in every arithmetic sequence whose terms do not have a
common factor, there are an infinite number of primes.
Mersenne numbers and Mersenne primes
Mersenne numbers
If 2n − 1 is prime, then n is prime, because if n had a prime factor p
then 2p − 1 would divide 2n − 1. If n = pq, then
2n − 1 = (2p − 1)(2pq − p + . . . + 23p + 22p + 2p + 1)
The converse, however, is not always true. If n is prime, 2n − 1 may
be either prime or composite.
primorial primes
Primorial primes are of the form p# + or - 1. George Pólya is reputed to
have replied to a pupil who asked how often the product of the first
n primes, plus 1, was itself prime, “There are many questions which
fools can ask that wise men cannot answer.” Gauss is supposed to
have made much the same answer to the question why did he not
tackle Fermat’s Last Theorem. (Golomb 1981)
It is not known if p# + 1 contains an infinity of primes, or even if an
infinity are composite.
Here one selects from random numbers the Nth one that corresponds to the answer. So for example for a prime P the Nth random number might be selected as prime or composite and this would continue for all higher values of P. So again there seems no reason why there would not be an infinite number of primes.
Siamese primes
Named by Beauregard and Suryanarayan, they are prime pairs of the
form n2 − 2 and n2 + 2. The sequence of pairs starts, 7-11, 79-83, 223-
227, 439-443, 1087-1091, 13687-13691, . . . (Beauregard and Suryanarayan
2001)