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More about prime numbers

Prime numbers play an important role in mathematics. In number theory, part of mathematics, prime numbers take a prominent place. Many mathematicians are looking for a solution or a proof for all kind of statements about prime numbers. Among this statements the Goldbach conjecture (that every even number greater than two is the sum of two prime numbers) and the
"prime k-tuple conjecture" that deals with patterns in consequtive prime numbers.
Then there is the "Riemann hypothesis" that deals with the distribution of prime numbers over the natural numbers.
This Riemann-Hypothesis is named as one of the seven "Millennium Problems"; problems that at the beginning of the millennium are defined as the most important not solved mathematical problems of this time. For the solution of each of this mathematical problems is a promised prize of one million dollars.

Some of the conjectures about prime numbers are centuries old; the availability of computers justifies the expectation that now sooner prove or solutions will be found for these conjectures. Because computers are an excellent tool for the study of prime numbers as evidenced by the book "Prime Numbers - A Computational Perspective", in which for many issues around primes, algorithms are described, in such a way that for this algorithms simply computer programs can be created.

So there is a program "Prime Number Research Program" (PrP) that can be downloaded here. This Windows program generates prime numbers up till 264; can amongst others count prime numers; and can make statistics of prime gaps. Furter the program can search for prime k-tuple clusters (also called prime constellations).

Description and user manual of the Prime Number Research Programma can you find here.

One of the enduring questions around prime numbers is: how many prime numbers are there less than or equal to a value x. Therefore, a special symbol is established: π(x); not to be confused with pi as ratio between circumference and diameter of a circle. So π(100) means: the number of prime numbers less than or equal to 100.
The maximum value of x for which π(x) can be determined, is continuously adjusted upwards. At this time (early 2015), the value π(1024) is set and by a second instance independently confirmed. This value is: 18,435,599,767,349,200,867,866. So many primes there are less than 1024. See the website about counting primes.

At that web site, a table is presented with numbers of primes less than an exponent of 10. Just because a computer usually works in the binary system, (so with numbers with exponents of 2), you would expect also a table with primes smaller than an exponent of 2 (which is equivalent to number of bits). Because this table is not found so far on the internet, a beginning of such a a table is made here. The table here presented goes up to and including 232. The challenge is to expand and complete this table to 248 and later may be to 264. The Windows program Prime Number Research Program can be of help.

Table with numbers with from one bit up to and including 16 bits:
(The single zero will be counted as a number)

Number with
number of bits
Amount of
different numbers:
Of this are
prime numbers:
Of this are
prime twins:(*)
1 bit 2 0 0
2 bits 2 2 0
3 bits 4 2 **
4 bits 8 2 2
5 bits 16 5 4
6 bits 32 7 4
7 bits 64 13 6
8 bits 128 23 14
9 bits 256 43 14
10 bits 512 75 24
11 bits 1,024 137 52
12 bits 2,048 255 90
13 bits 4,096 464 140
14 bits 8,192 872 226
15 bits 16,384 1,612 430
16 bits 32,768 3,030 710
Total 65,536 6,542 1,716

(*) Each twin prime is counted for two.
(**) 3,5,7 form a prime triplet and is not counted as twin primes.

The following table contains numbers with from 17 bits up to and including 32 bits:

Number with
number of bits
Amount of
different numbers:
Of this are
prime numbers:
Of this are
prime twins(*)
1-16 bits 65,536 6,542 1,716
17 bits 65,536 5,709 1,332
18 bits 131,072 10,749 2,306
19 bits 262,144 20,390 4,142
20 bits 524,288 38,635 7,570
21 bits 1,048,576 73,586 13,930
22 bits 2,097,152 140,336 24,990
23 bits 4,194,304 268,216 45,286
24 bits 8,388,608 513,708 83,216
25 bits 16,777,216 985,818 152,742
26 bits 33,554,432 1,894,120 281,888
27 bits 67,108,864 3,645,744 523,504
28 bits 134,217,728 7,027,290 969,936
29 bits 268,435,456 13,561,907 1,809,598
30 bits 536,870,912 26,207,278 3,378,954
31 bits 1,073,741,824 50,697,537 6,320,226
32 bits 2,147,483,648 98,182,656 11,857,808
Total 4,294,967,296 203,280,221 25,479,144

(*) Each twin prime is counted for two.

The following table contains numbers with from 33 bits up to and including 48 bits:
(This table needs to be completed).

Number with
number of bits
Amount of
different numbers:
Of this are
prime numbers:
Of this are
twin primes(*)
1-32 bit 4,294,967,296 203,280,221 25,479,144
33 bits 4,294,967,296 190,335,585 22,278,142
34 bits 8,589,934,592 369,323,305 41,941,564
35 bits 17,179,869,184 717,267,168 79,070,162
36 bits 34,359,738,368 1,394,192,236 149,395,490
37 bits 68,719,476,736 2,712,103,833 282,684,980
38 bits 137,438,953,472 5,279,763,824 535,624,524
39 bits 274,877,906,94410,285,641,7781,016,388,188
40 bits 549,755,813,88820,051,180,8461,931,246,464
41 bits 1,099,511,627,77639,113,482,6403,674,295,434
42 bits 2,199,023,255,55276,344,462,7976,999,452,962
43 bits 4,398,046,511,104149,100,679,00413,348,502,746
44 bits 8,796,093,022,208291,354,668,49525,485,058,834
45 bits 17,592,186,044,416569,630,404,45048,707,498,526
46 bits 35,184,372,088,8321,114,251,967,76793,184,432,092
47 bits 70,368,744,177,6642,180,634,225,768 178,448,452,644
48 bits140,737,488,355,328
Total 281,474,976,710,656

(*) Each twin prime is counted for two.

december 31, 2020