Prime numbers are simple enough in their basic make up and definition, but there are some unknown aspects of prime numbers that even until today defy the best brains of the best mathematicians in the world of today.
Primes numbers are deceptively simple. They are defined as being any whole number that is divisible only by itself and the number one.
A mathematician would put it more complexly stated maybe something like this:
A prime number is a natural number that is an integer greater than one if its only positive divisors, ( called factors) are one and itself.
An integer can be sometimes be divided by numbers that are known as prime devisors. The prime devisors of the number 15, for example are the numbers 3 and 5, because these devisors are prime numbers. The number 8 can be divided by the numbers 2 and 4, but these are not prime devisors.
Prime numbers are known as the building blocks of all positive numbers. Every positive number is a product of prime numbers in one and only one way. For example the number 100. It’s product using only prime numbers is like this: 5 x 5 x 2 x 2. Prime numbers can generate all the other integers in this way.
The number one was once also included within the field of prime numbers, but it is no longer included now because of the more stricter definition of what a prime number is. The number one is somewhat unique in itself and doesn’t follow the modern definition of a prime number. Every number has a unique factorization of itself into primes called prime devisors above. The number one doesn’t meet this criteria. It is only itself (1 x 1 = 1), and can only be factored into itself. It doesn’t have any other prime devisors as such.
It was shown way back in 300 BC by the Greek mathematician Euclid that there are an infinite number of prime numbers and that they are all spaced apart in a funny way. The spacing is irregular and some times they can be far apart, and sometimes there can be a pair of primes that are very close together. For example the prime numbers 41 and 43, and 101 and 103, are instances of being a pair of such primes. Between the numbers 109 and 127 though, there is a large gap where there is only one other prime number, the number 113.
These pairs of primes are also known as twin primes. They differ from the preceding prime number only by the number 2. Strangely enough this occurrence of twin primes is also an infinite occurrence within the field of prime numbers and doesn’t stop from ever happening at some stage after millions of primes have already been identified.
The largest know twin prime number discovered up until now has 58711 digits to it! Examples earlier than that are 881 and 883, or 599 and 601.
There has never been a method found to determine if a number is a prime number or not except by using a cumbersome method that was discovered by the Greek mathematician Eratosthenes who was quite famous for his work on prime numbers and whom the “Sieve of Eratosthenes” is named after.
Eratosthenes, (275 to 194 BC) devised a type of sieve or strainer method to discover prime numbers. His method was simply to take a number and then to start to divide it by all the other numbers starting from the number 2. You keep going until you have reached the number which is the square root of your chosen number, and by which time you will have already now established whether the number is prime or not.
When applied to a grid of numbers Eratosthenes’s method separates out all of the other composite numbers leaving only the prime numbers all behind. A composite number is any positive number having another positive divisor other than being only one or itself.
For example the number 799, is it a prime number?
You divide this number 799, by all of the other numbers beginning from the number 3, because all numbers divisible by two are of course not prime numbers. Then divide by the other odd numbers until you reach the number 17. You will find that 17 x 47 is 799, and so 799 is not a prime number.
How about the number 499?
Start again from the number 3 and work yourself through the other odd numbers until you reach the number 21. This is far enough as you will begin to see that you are only just repeating and going in reverse again after that. If you take the square root of your suspected prime number, you never need to go past that number in using this method to determine if your number is a prime number or not.
A square root of a number is a number that when multiplied by itself is equal to that given number. For example the square root of 9 is the number 3.
The square root of 499 is 22.34. By the time we reach 21 in this method of division into the test number 499, we already know that indeed the number 499 is a prime number. The square root of a prime number will never be a whole number.
This method of finding prime numbers is time consuming and cumbersome, but not of course for the large computers on which mathematicians are still working frantically to try to find the next largest prime number to be discovered by them.
An example of a large prime number is the number, 950200117 and 950200217 is the next one after that.
There is an interesting web site called www.prime-numbers.org in which you can browse all of the prime numbers smaller than 10,000,000,000, and you can also plug in any number lower than that to check if it is really a prime number or not.
Funnily enough this method of Eratosthenes is still the most exactly efficient way of finding all of the very small prime numbers, (for those numbers less than 1,000,000). The largest prime numbers are now found using group theory, but no definite formula has so far been discovered to ascertain for certain if a number is prime or not, except for this hard and time consuming method of number crunching.
Other forms of sieves have been developed since then. One such sieve is known as the sieve of Atkin and which is an algorithm or sequence of instructions which when followed will find prime numbers up until a certain given number.
There are other forms prime numbers for example, primorial primes and factorial primes, but this short introductory article is primarily only about simple prime numbers.
A provable prime number is an integer that has been proven to be a prime number using a primality proving algorithm. A probable prime is a number that only has a high possibility of being a prime number based on the results obtained from the use of a probabilistic algorithm. A primality test is an algorithm used for determining if a number that is fed into it is a prime number or not.
In 1984 the mathematician Samuel Yates defined a titanic prime number to be a prime number with at least 1000 digits.
When he first coined this name there were only 110 such prime numbers known to his fellow mathematicians in existence. At the time of my writing this article in 2009, there are now over a thousand times that many titanic prime numbers recorded and recognised to be prime numbers. This number will of course only ever continue to grow.
The number of prime numbers was proven to be infinite by Euclid, (the Greek Mathematician from 300 BC mentioned briefly above), and so we expect to find even larger and larger prime numbers forever into our future. The first billion digit prime number is out there somewhere only waiting for someone like you to find it!
Maths has often been linked to numerology and to religion and the prime number can also be interpreted to have spiritual significances in this way. The early mathematicians were also religious mystics in their own ways.
God is seen to be one substance, and there are an infinite number of souls or prime vehicles created for his love to reach into and to utilise. Spiritually a prime number is showing this principal of the soul being linked primarily only to the one, or to God.
The ancient Greek mathematicians looked for deeper spiritual meaning within their understanding of the nature and function of mathematics.
These Greek mathematicians defined a perfect number to be a number in which the sum of its divisors or factors equalled itself. Number 6 is the first perfect number. It’s factors are 1, 2 and 3 and these numbers sum up to itself, the number 6. The next two perfect numbers are the numbers 28 and 496. Euclid deduced that a perfect number was somehow related to a prime number. He came up with a formula based on the use of prime numbers to generate his perfect numbers. Euclid was a great believer and follower of numerology.
Even today Euclid’s maths cannot be improved upon. Mathematicians even now cannot predict where a perfect number occurs between two randomly selected prime numbers.
What adds a real mystical touch of authenticity or mystery to all of this is the fact that Greek arithmetic was not based on the same ten based number system as ours is now, and yet the same rules applied to the existence of prime or perfect numbers. The base being used was not an influencing factor apparently.
Amazing stuff and I hope I have whetted your appetite to further probe into this field of prime numbers while still in your prime!
A prime number or soul is uniquely only itself, and so it cannot be determined by any other souls except for itself and God, (read one and itself) and this is why maths will never discover the complete method of locating a prime number. It keeps its independence and freedom to itself.
A prime number is itself said to be resting within God representing the number one, as is soul. There is no known and certain method to ascertain whether a number is a prime number or not except for meeting that soul or number within itself or yourself as a prime number linked to God or back to the number one. A soul meeting another prime number and soul combines a message of love and this equates in maths terms to a prime number being the prime and basic force of all other numbers and as is your soul.
The mind wants formulas, but the heart only loves. You cannot go into your mind to find what isn’t contained there. Each soul and each prime number is unique in its expression, and cannot be worked out by another method except for being found uniquely as itself unless you find your true soul mate and which perhaps is only really a twin prime!
