Primes are infintie.[ Proof by contradiction] and background.
Primes are infintie.[ Proof by contradiction] and background.
ghost | 5551 Reads | A background: Numbers, or more specifically, Natural numbers[ Positive intigers-whole numbers above zero] have factors. Factors are number with which these intigers can be divided by, for example, 6 has a factor of 2 and 3, as 2*3=6. Prime numbers are special in that they only have 2 different factors, 1 and themselves.
The first 5 prime numbers are: 2, 3, 5, 7, 11…. Now a few sharp but if you read the rule defining primes above they must have different factors, so 1 is not prime as 1 and its self are the same number.
Every non prime[ compsite is the mathematical term] number has prime factors, for example 56 has the prime factors 2, 2, 2, 7. Every non-prime number may be written in this form, this leads to the conclusion, a number is prime because it cannot be made by multiplying to or more of the prime numbers smaller than it.[ remmeber this.] The encryption system based on primes utilises this concept, if you get prime numbers big enough and multiply them then you will have a very hard number to factorise. This only wokrs with primes because if you used a non-prime number then every factor of them would also be a factor of the final number. Okay, so now to the proof…
This is a proof by contradiction, this means we make aassumption and then disprove it. This is a very simple and old proof.
Assume: THERE ARE A FINITE AMOUNT OF PRIMES.
X is the final prime. Now, n=235711*…X Now n itself now has factors however, if we add one to it then if we divide it by any number we will end up with atleast a remnder of 1. Think of it this way the highest common factor of any two consecutive numbers must be one. say we have n and n+6, then the differnce is 6 which means the possible facotors are 2,3 or 6. However with n and n+1 it can only be one, also as no other number below it are factors, then the only remaining facotr is itself therefore n+1 is prime, ASIDE| 2 is prime 3 is prime, 23+1=7, 7 is prime, do it as much as you like.
This can be continued over and over to prove that they are infinite.
While this proof is very simple and little knowledge of algerba is needed it takes a good meausre of sound mathematics. To quote Bohr roughly: " If one does not feel dizzy when contemplating the menaing of Plank constant then one does not know what onesis talking about." It takes a good understanding to cause problems when thinking about this, and even better understanding to actually assure your self of it.
This means even as computer evolve to get better at factorising numbers, computer get better at finding primes! Some 20 years ago athe largest primes where about 100 digits long with modern ones reahcing around a million digits in legnth.
A Mersenne prime is a prime which can be expressed in the form 2(p)-1. That is one less than a power of two, there is a link with Mersenne primes and perfect numbers[ who's proper factors add up to equal the number] by the rule if M is prime then M(M+1)/2 is a perfect number, htis link is what led to such interest into Mersenne primes, there are several projects searching for such primes. for example GIMPS with the name aim of finding a million digit Mersenne prime.
"God may not play dice with the universe, but something strange is going on with the prime numbers."  Paul Erdős
