prime numbers: facts and discoveries. MIND BOGGLING!!!!. Facts:. The largest known prime number. The largest known prime number is 2 57,885,161 − 1 , a number with 17,425,170 digits . How many prime numbers are there?
prime numbers: facts anddiscoveries
The largest known prime number
The percentage decreases with the increase in numbers
You can drop off any amount of digits from a side and the resulting number will remain prime
19 -> 31 -> 73 ->97
131 -> 373 -> 797 -> 919
157 -> 359 -> 751 -> 953
821 -> 823 -> 827 -> 829
The mathematics behind this is rather simple.1. Let p be a prime number, p > 3.2. Squaring gives: p^2.3. Adding 14 gives: p^2 + 144. Taking it modulo 12 gives: (p^2 + 14) mod 12
We want to show that: (p^2 + 14) mod 12 = 3This is equivalent to: p^2 – 1 is divisible by 12.That is: (p-1)(p+1) is divisible by 12.
For a number to be divisible by twelve, it has to be divisible both by 3 and by 4. We know that, out of p-1, p and p+1, one of them must bedivisible by 3; and it can’t be p, because p is prime and greater than 3. Thus, either p-1 or p+1 is divisible by 3, and so their product is also: (p-1)(p+1) is divisible by 3.
Now, since p is a prime greater than 3, we know that it is odd. Therefore, both p-1 and p+1 are even numbers. The product of two even numbers is divisible by 4, so: (p-1)(p+1) is divisible by 4.
Combining this with the above, we get that: (p-1)(p+1) is divisible by 12.And hence: (p^2 + 14) mod 12 = 3
Prime numbers are not all that useful but they are interesting. Here is an interesting fact that you may like to think about.
The astronomer Carl Sagan wanted to broadcast prime numbers to the Universe as a signal to alien life forms that we exist & are intelligent.
bibliography divisible both by 3 and by 4. We know that, out of p-1, p and p+1, one of them must be
Efforts by: divisible both by 3 and by 4. We know that, out of p-1, p and p+1, one of them must be