Prime numbers (or Primes) are natural numbers, greater than 1, that are only divisible by itself and by one. The first prime number is 2. Then follows 3, 5, 7, 11 etc.
Prime numbers occur relatively frequent. Of the first 100 natural numbers 25 are prime. So that is 25%. As numbers are becoming larger the percentage of primes decreases. So for the first thousand natural numbers 16% is prime, but for the first one million numbers the percentage is only 8. It is proven that there are infinitely many prime numbers.
There is only one even prime number: that is 2. All remaining prime numbers are odd. Whenever two odd primes are next to each other with only one even number in between, then they are called twin prime. For example 11 and 13 is a twin prime; so is 17 and 19. There is also a prime triplet; three odd primes with only one even number between them. There is only one real prime triplet: 3, 5, 7. Three consecutive primes with a difference of two and four, or with a difference of four and two, is also called a prime triplet. There are many of them.
There are four primes with one digit: 2, 3, 5, 7. There are 21 primes with two digits:
11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 and 97.
The billionth prime number is: 22.801.763.489. This prime number you can find with the Windows program "Billions of Primes" (See next page)
List of the amount of prime numbers with certain digits:
| Number of | Amount of different numbers | Of this are prime numbers: | Of this are twin primes:(*) |
|---|---|---|---|
| 1 digit | 10 | 4 | One prime triplet(**) |
| 2 digits | 90 | 21 | 12 |
| 3 digits | 900 | 143 | 54 |
| 4 digits | 9.000 | 1.061 | 340 |
| 5 digits | 90.000 | 8.363 | 2038 |
| 6 digits | 900.000 | 68.906 | 13.890 |
| 7 digits | 9.000.000 | 586.081 | 101.622 |
| 8 digits | 90.000.000 | 5.096.876 | 762.664 |
| 9 digits | 900.000.000 | 45.086.079 | 5.968.388 |
| 10 digits | 9.000.000.000 | 404.204.977 | 47.976.346 |
| 11 digits | 90.000.000.000 | 3.663.002.302 | 393.926.738 |
| 12 digits | 900.000.000.000 | 33.489.857.205 | 3.292.418.344 |
| Total | 1.000.000.000.000 | 37.607.912.018 | 3.741.170.436 |
(*) Eeach twin prime is counted for two.
(**) 3,5 and 7 form a prime triplet and are not counted in number of twin primes
As you can see there are a lot of prime numbers, especially in the range of numbers with more digits. There are sufficient 12-digit prime numbers to assign every world citizen a unique prime-number. (That could be a Global Social Security Number). Even every human twin could get a unique 12-digit twin prime!
There is a computer program that can show all prime numbers up to 12 digits. That is the Windows program Billions of Primes and that can be downloaded here for free. In that program you can also find the billionth prime number.
Natural numbers are the numbers 1, 2, 3, 4, 5 etc. All natural numbers can be factorized
into prime numbers ór are prime numbers themselfs. In this way we can consider prime numbers as
the building blocks of natural numbers. Small numbers, like 36, we can factorize into primes without tools:
36 = 2 x 2 x 3 x 3.
But large numbers cannot be factorized into primes without tools.
For example 2,109,662,887 = 35,381 x 59,627.
The numbers 35,381 and 59,627 are prime numbers. You cannot do that easy without a tool.
There is a Windows program
Factoring into Primes that can factorize numbers up to 18 decimal digits within a few
seconds.
Numbers in the form of (2n-1)
are often prime numbers for values of n that are primes themselfs.
A French monk was the discoverer of this phenomena. In 1644 he published an article bout this item.
The name of the monk was Marin Mersenne, and prime numbers of this form are given the name Mersenne Prime Number.
Since the publication mathematicians have ever been looking for new (larger) Mersenne Primes.
This quest for larger Prime Numbers
accelerated when computers became available.
In 1996 Georg Woltman started a project to use private Personal Computers to help discover
new Mersenne Primes by providing programs and data via internet.
Privately owned internet connected computers can cooperate in the project, which has got the name
GIMPS (the Great Internet Mersenne Prime Search) and can be found via
www.mersenne.org .
All Mersenne Primes known until now (january 2018) can be shown via the Windows Mersenne Prime Program.
The largest known Prime Number is (as of this moment january 2018) a Mersenne Prime of more than 23 million digits. It is found by the GIMPS project. When this number is printed it results in a pile of more than six thousand A4 sheets, full of just decimal digits!
All Prime Numbers less than one trillion can be found by the Windows Program Billions of Primes. Larger primes can be generated by the Windows Program Generate Prime Numbers . These larger prime numbers can be used for instance by cryptography, to generate cryptographic keys. The size of prime numbers in use by cryptographic systems is usualy expressed in number of bits. The generate prime number program can produce primes from 33 bits up to 1,000 bits.
The distance between two consecutive primes is called prime gap. The value of a prime gap is the difference between the larger and the smaller prime of two consecutive primes. Apart from the first prime gap (between the first and the second prime number, ie between 2 and 3), which has the value 1, all of the other gaps are an even number. There exists 37,607,912,018 prime numbers bellow one trillion (1,000,000,000,000) and so the number of prime gaps is one less. Of these 37,607,912,017 gaps the smallest even gap has a value of 2 and the largest gap has a value of 540. All even numbers up to 480 occur as prime gap. More than 9% of the gaps has a size of 6, making this the most common prime gap size. (This is true at least for all prime numbers up to a trillion). The following statistic is a listing of all the prime gaps for all prime numbers up to a trillion.
A Prime Number can be called "lonely" if it is far away from the nearest smaller or larger prime number.
As Prime Numbers become larger and larger, the gaps between primes also increases in size. So also
the "loneliness" increases.
However Prime Numbers are not evenly distributed over the natural numbers and thus also the prime gaps are
of arbitrary sizes.
Up to a certain number there is a "most lonely" Prime Number. But if we look further we will meet a new
Prime Number that can be "more lonely" than the previous one.
In this report you can find the most lonely prime numbers
up to one trillion.
About Twin Primes you can hardly say that they are lonely, because they have each other; there is only
one (even) number between the two of a Twin Prime.
However, when we consider a twin prime as a whole you can speak of a degree of being lonely,
namely how far removed the pair is from the next smaller prime number or the next larger prime number.
Like for most lonely Prime Numbers, there can be created a list of
most lonely twin primes
In his book "The Solitude of Prime Numbers", Paolo Giordano mentions two prime numbers:
2,760,889,966,649 and 2,760,889,966,651. They form a twin prime.
The primes are close to each other but are in fact for ever separated by an even number. That is what the title
of this book expresses.
We can, however, in a different way look at the lonelyless of this primes, namely we can look at
how far this twin prime is separated from other prime numbers.
The distance of this twin prime to the next smaller prime number is 46 and the distance to the next larger prime
number is only 6.
Based on the aforementioned report "most lonely prime numbers" you would expect that these distances
could be much greater.
Not too far from the twin prime mentioned in the book is a twin prime significantly
more "lonely": 2,760,889,963,619 and 2,760,889,963,621. The distance from this twin prime
to the next smaller and the next larger prime is respectively 66 and 60.