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 | 274,877,906,944 | 10,285,641,778 | 1,016,388,188 |
| 40 bits | 549,755,813,888 | 20,051,180,846 | 1,931,246,464 |
| 41 bits | 1,099,511,627,776 | 39,113,482,640 | 3,674,295,434 |
| 42 bits | 2,199,023,255,552 | 76,344,462,797 | 6,999,452,962 |
| 43 bits | 4,398,046,511,104 | 149,100,679,004 | 13,348,502,746 |
| 44 bits | 8,796,093,022,208 | 291,354,668,495 | 25,485,058,834 |
| 45 bits | 17,592,186,044,416 | 569,630,404,450 | 48,707,498,526 |
| 46 bits | 35,184,372,088,832 | 1,114,251,967,767 | 93,184,432,092 |
| 47 bits | 70,368,744,177,664 | 2,180,634,225,768 | 178,448,452,644 |
| 48 bits | 140,737,488,355,328 | ||
| Total | 281,474,976,710,656 |
(*) Each twin prime is counted for two.
december 31, 2020