In school, we learn that prime numbers are numbers which are only divisible by 1 and itself. Hence, the prime numbers are 2, 3, 5, 7, 11, ... the list goes on and on. You might wonder why 1 isn't a prime number since according to the definition, 1 should be! However, many mathematicians dismiss 1 as a prime number and start with 2 in the series of prime numbers as it enables theorems to be elegantly stated. For example, prime factorization of 24 is uniquely set as 2^3 X 3^1. If we allow 1 into our list of prime numbers, prime factorization of 24 could be 1 X 2^3 X 3^1 or even 1^234567876543234567 X 2^3 X 3^1! Moreover, the definition of prime numbers states that they have to be divisible by 1 and itself. Maybe we should further add in a side note that 1 not equals to itself. Mathematicians have weird logical reasonings sometimes!
Moving on, prime numbers are very much to Mathematics what atoms are to Chemistry. Like atoms, which are building blocks of matter, all numbers can be expressed in the form of prime factorization as shown above. There is a name for this too! It's called the 'prime-number decomposition theorem'. This theorem states that every whole number greater than 1 can be written by multiplying prime numbers in exactly one way.
Having said so much, you might like to know that the greatest prime number as of 28/09/2008 is 2^43112609 - 1! That is close to 13 million digits long! In fact, the list of prime numbers goes on and on to infinity as proven by Euclid. In his book Elements (Book 9, Proposition 20), he stated that 'Prime numbers are more than any assigned multitude of prime numbers' and proceeded to write his proof down. Though quite short, I shall not bore you with sleep inducing proofs like this. If you're interested, you can read the footnote at the bottom.
The number 666, commonly known as the 'number of the beast' in the biblical book of Revelations, has some unexpected properties regarding prime numbers. It is the sum of the squares of the first 7 primes: 666 = 2^2 + 3^2 + 5^2 + 11^2 +13^2 + 17^2. Impressed? You might want to know that 666 is also the sum of palindromic cubes too: 666 = 1^3 +2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 5^3 + 4^3 + 3^3 + 2^3 + 1^3. And note: the keystone 6^3 in the centre is shorthand for 6 X 6 X 6! Need I say more?
Well. The amount of knowledge of prime numbers that is known is already so much, but there are still many unknown areas about it waiting to be explored. A famous one would be the 'Goldbach Conjecture'. Christian Goldbach conjectured that 'Every even number greater than 2 is the sum of two prime numbers.' For instance, 42 is an even number and is the sum of 5 and 37, both prime numbers. It can even be 11 + 31, 13 + 29 or 19 + 23. The conjecture is true for many numbers, but it has never been proved in general. Progress has been made towards the proof, and many mathematicians feel that the proof isn't far off. Chinese mathematician Chen Jingrun made a great step by proving his theorem that 'Every sufficiently large even number can be written as the sum of two primes of the sum of a prime and a semi-prime (a number which is the multiplication of two primes).' I hope I get to read the proof one day - not in chinese though because my chinese is horrible!
Ending this long entry about prime numbers, prime numbers can be broken down too, much like atoms into smaller units like quarks. In the realm of complex numbers, prime numbers like 5 can be expressed as a product of two numbers. For example, 5 = (1 - 2i) X (1 + 2i), where i is the square root of -1 of the imaginary number system. As a product of two Gaussian integers, prime numbers are not as unbreakable as was once supposed!
I hope I enlightened you somewhat on this entry about prime numbers and sparked some interest in you wanting to know more about the wonders of Mathematics. Though intrinsically basic, prime numbers offers stiff challenges for those who delve deeper into its roots. Feel free to comment on this entry. Cheers!
Footnote. Euclid's Elements (Book 9, Proposition 20)
Suppose that P is the largest prime, and consider the number N = (2 X 3 X 5 X ... X P) + 1. Either N is prime or it is not. If N is prime, we have produced a prime greater than P, which is a contradiction to our supposition. If N is not a prime, it must be divisible by some prime, say p, which is one of 2, 3, 5, ..., P. This means that p divides N - (2 X 3 X 5 X ... X P). But this number is equals to 1 and so p divides 1. This cannot be since all primes are greater than 1. Thus, whatever the nature of N, we arrive at a contradiction. Our original assumption of there being a largest prime P is therefore false. Conclusion: the number of prime is limitless.
Disclaimer: This entry is written based on the knowledge of the book '50 mathematical ideas you really need to know' by Tony Crilly.
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