=> 2, 3, 5 .......... n must be the list of ALL prime numbers. any number not in the list is not prime, n is the largest no. in the list.
=> (2*3*5*.......*n + 1) is not divisible by any number (except 1 and itself)
=> (2*3*5*.......*n + 1) is also a prime number
but (2*3*5*.......*n + 1) > n => contradiction (we assumed n is the largest prime)
=> there is no largest prime number DONE!
(remember that if a number is not prime <=> number is divisible by some prime number(s))
(2 power any prime number) -1 gives one group of prime numbers. e.g. 3, 7, 31 etc.
However, there is another group of prime numbers too which can't be obtained by (2 power any prime number) -1. e.g. 13, 19 etc.
But if u want to prove that there is no largest prime number by using (2 power any prime number) -1 form, you are, sorry to say that, wrong.
As in that case largest prime number is 232,582,657-1 (discovered in sept, 06, as far as i remember)
that means 232,582,657-1 is the largest prime number known of (2 power any prime number) -1 type.
Please see internet to confirm.
Correct me IF i m somewhere wrong. I may be wrong as this topic is very very controversial among mathematicians. Who knows u will find a bigger prime number than the existing one and will be the next Ramanujan :)
But Yes, there is no largest prime number and that is possible becoz they may not be of (2 power any prime number) -1 type.
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