Prime number is a number that does not divide perfectly into other number. When I said perfectly — I mean with no remainder. Its only factors are 1 and itself. Prime numbers form other numbers that are known as composite numbers i.e. prime numbers are factors of composite numbers.
The process of seeing how numbers can be broken down relates to their prime factors. For example, 1000 divided by 15 can be estimated by breaking down 15 into its prime factors 3 and 5, so we can go straight dividing 250 by 3 for the answer, which makes it a lot easier than doing direct division of 1000 with 15 .
Sequence of prime numbers
Except for 2, all other primes are odd numbers. However, this does not mean all odd numbers are primes. What can be more meaningful is actually to know which odd numbers that are primes —by knowing how an odd number is factorised by its primes. When we know that, we can work out the sequence of prime numbers by considering smaller prime numbers that are factors of odd numbers, and excluding these composite odd numbers from our list.
We could also consider a sequence of odd numbers in tens with a specific last digit (or residue class of mod 10)— eg we can start with 3, 13, 23, 33, 43, 53, 63, 73, 83 and 93. It is obvious to see that odd numbers where the second digit ended with 3 is not necessarily a prime number if the first digit can be divided by 3.
So, is this a rule that work for all other consideration in this fashion?
5 for instance, is the only prime number as other subsequent numbers of the sequence (in this fashion) are all divisible by 5.
However, sequence of 7 works pretty much similar to 3, but there is an exception for the number where the first digit is 2 i.e. 27. But this exception can also be seen in other odd numbers with the first digit of 2 eg 21.
The sequence of prime numbers is actually difficult to determine.
But, can we say that numbers with certain end digits, in this case, 1, 3, 7 and 9 could be primes following a certain pattern?
Recently, I came across the Archimedean spiral pattern relating to the distribution of prime numbers that relates to rational approximation of pi — all by playing around plotting numbers on polar coordinates. I really recommend seeing the video here on YouTube: https://www.youtube.com/watch?v=EK32jo7i5LQ
Although it’s really easy to figure out that as long as the counting of numbers goes on there would always be the next prime number, proving for infinite sequence of primes has always been the interest of many great Mathematicians and scholars.
Infinitude of primes by contradiction
A common way to show proof for infinity sequence of prime numbers is using contradiction statement such as the Ribenboim’s statement.
Basically, this takes into consideration of two important representations.
First, N which is a composite number, in this case, where it is a product of other prime numbers:
Secondly, we need to make an assumption contradicting to infinite sequence of prime numbers. In this case, supposed there are finite prime numbers that exist in which there is a prime number p which is larger than all the prime factors within N.
In this case, we will consider the statement
Some of us may wonder why we use the statement with +1, and not +2, or +3 or other numbers. This is due to the fact that 1 is the only option we can use here that is not associated to prime.
(Remember: we need to find a contradiction for assumption made for a finite sequence of primes.)
The argument for contradiction is that for any Q divided by p, it would be impossible to get one of the prime numbers already in the list:
as there would always be consideration for the term:
So, in conclusion, there would always exist a p that is a prime number that is not already in the list. This means the list of primes goes on into infinity.
Ever thought of how this endless list of prime numbers can be manipulated?
The growing sequence of primes means it purposefully could serve for the uncertain encryption keys (composite to primes) which already is quite impossible for computer to solve.
Secure banking transactions use encryption of public and private keys with composite and prime numbers. Imagine a 32-bit number being used — 2,146,654,199 — imagine how impossible of finding the pair of primes associated to this particular number. (Try this if you can!)
This is not even considering the fact that bank encryption default is set at 2096 bits or even at 4096 bits for maximum security!
Why is it that sequencing the next and next prime numbers important?
What is it for really?
Perhaps the only reason is that it tells us where we are at the moment with the technology — i.e the technology to derive the largest prime number and the technology used for its verification.