
On. Feb. 12, 2011, I wrote a post on prime numbers: Prime Numbers, Large Numbers. This topic still fascinates me.
A prime number is a whole number that can only be divided by 1 and by itself. Or put differently, prime numbers cannot be divided by any other whole number without leaving a fraction. For example, the four prime numbers below ten are: 2, 3, 5, 7. Beyond that, no even number can be a prime. Nor can any number ending on 5, since all such numbers can be divided by 5.
One fascinating aspect of prime numbers is that they seem to obey no other law than that of chance, and that nobody can predict what the next higher - and not yet discovered - prime number will be!
Back in 2009, the largest prime number ever discovered (by the Greater Internet Mersenne Prime Search Research Project) was: 2 to the power 43,112,609 minus 1. This is a number with almost 13 million decimal digits.
Since then, the same researchers have gone much further. As of early 2014, the largest known prime is 2 to the power 57,885,161 minus 1. This number has nearly 17 and a half million digits. To write it out would require 6,000 pages, or twenty 300-page long books. For more information about prime numbers, see: prime numbers
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The way to find out whether a number is a prime:
First, as I just told you, no even number, nor any number ending on 5, can be a prime. So the only candidates to be prime numbers are those that end on 1,3,7 or 9. This already reduces your task by 60%.
But then, you have to factorize: That is, when you try to determine whether a number is a prime or not, you must “decompose” it and find its factors that are prime (indivisible). For example, take 87: It is divisible by two prime numbers: 3 (and 29). Therefore 87 is not a prime.
Now try to factorize 89: is it divisible by 3? No. By 7? No. By 9? No. Etc. 89 turns out to be a prime.
Here are the 25 primes under 100: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97.
As you get into large numbers, factorization becomes tediously time-consuming. The good news is that there are many prime number calculators on the Internet, for example: calculator, which you can use up to nearly four and a half billion.
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Here is one of the most fascinating aspects of prime numbers: There is an INFINITE amount of them, but at the same time, they become more and more rare as you move up the sequence of natural numbers. This is something you suspect right away, even just looking at the first few hundred numbers: For example, here is how many prime numbers there are under 500:
from 1 to 100: 25
101 to 200: 21
201 to 300: 16
301 to 400: 17
401 to 500: 16
The diminishing number of primes is called the Prime Number Theorem (PNT).
Mathematicians have known this for hundreds of years. For example, Carl Friedrich Gauss - of the famous Gaussian (normal) distribution - studied and confirmed this theorem in 1791, at the age of 14.
Mathematicians say that “the distribution of primes is ASYMPTOTIC.” This means that the curve approaches zero as it tends towards infinity. In other words, it never touches zero, and its slope is curvilinear, declining ever more slowly. See: https://www.google.com/#q=asymptotic+curve
I just wanted to SEE the asymptotic distribution of prime numbers with my own eyes, so I examined some tables showing the density of primes as you go up the sequence of natural numbers. Here is what I found:
below 10, there are 4 prime numbers = 40 per 100
below 100: 25 = 25 per 100
below 1,000: 138 = 14 per 100
below 10,000: 1,229 = 12 per 100
below 100,000: 9,592 = 10 per 100
below 1 million: 78,498 = 8 per 100
below 10 million : 664,579 = 7 per 100
below 100 million: 5,761,455 = 6 per 100
below 1 billion: 50,847,534 = 5 per 100
below 10 billion: 455,052,511 = 4.6 per 100
Etc.
See: questions about prime numbers
So it’s clear that prime numbers become more rare as you go up the series of natural numbers. Also, the DECLINE in the frequency of prime numbers slows down.
Mathematicians say that the frequency reaches zero in infinity or at the “limit.” In other words, there NEVER comes a number, no matter how unimaginably large, beyond which there are zero primes, or only one, or only a finite amount of them, even though they are spaced further and further apart, which is what makes finding them increasingly difficult, even with the most powerful computers.
I find it incredibly weird that there are immense numbers out there which are indivisible, in other words, quantities of just ONE piece, without ANY component parts.
© Tom Kando 2014
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