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Primes numbers are divisible only by 1 and by themselves. Is it possible to to predict where primes will appear in the series of natural numbers 1,2,3...?
Some people believe that there are hidden patterns in the series of prime numbers, which still need to be discovered, and which will allow us to calculate where prime numbers appear in the series.
If the only divisors of natural numbers were 0,1,2 and 3, and if we considered 5 and numbers bigger than 5 not as a divisors, but only as natural numbers to be divided, we would have a regular pattern in primes. All prime numbers are odd (with the exception of 2), because every even number is divisible by 2 (therefore has at least 3 factors (1,2 and itself). If you divided the series of odd numbers by 3,  one third of the odd numbers would be divisible by 3. We could find the primes by a process of exclusion: we would know that only the odd numbers include the primes, and that every odd number which is not a multiple of 3 would be a prime (apart from 2), so it would be easy to calculate primes. For example, the first primes would be 2,3,5,7,11,13,17,19,23,25,29, 31,35,37,41,43,47,49,53,55 etc.
To find the primes, we would start from 5, and add first 2, then 4 (alternately), and we would find the next prime number.
However, when we use number 5 as a prime and as a new divisor (together with 3), we  generate irregularity and randomness in the previous series, as multiples of 5 will make composite (and not primes any more) new numbers from the previous series, like 25, 35, 55 etc. As a consequence, we would have to remove these numbers from the initial series. It would be like finding random holes in a road where holes appeared regularly every 2 odd numbers.

When we introduce 7 as a prime divisor, we will find more odd composite numbers that are not divisible by 3 or 5, but are divisible by 7, like 49, 77,91 etc. We would have to remove these numbers as well, and more random holes would appear on the road.

As we use more and more primes as divisors (11,13, 17, 19 etc.), we will find new odd numbers in our initial series that are divisible only by these new primes, for example, the odd number 209 (which is divisible only by 11, and not any previous prime); the odd number 247 (which is divisible only by 13 etc). Therefore we will have to remove these numbers from our initial list of primes, and more and more numbers will have to be removed, as as we keep finding random new primes.

Euclid demonstrated that there are  infinite primes.  Therefore there are infinite new unique divisors of infinite new odd numbers, which causes infinite randomness in the series.

I suspect that a finite number of primes is not enough to generate all the unique prime factors. For any given prime number p1, you can find a composite odd number n which equals p1*p1, and is divisible only by p, and is not divisible by any other prime smaller than p (and it's the square of a prime, having as factors 1, itself, and the prime p).

In other words, you don't know how many odd numbers you will have to remove from our initial series, therefore you can't predict where new holes will appear in the initial series. In fact, they proved that there are a number of consecutive holes (consecutive composite numbers) in the series of natural numbers as big as any given natural number, by using an proof similar to the one used by Euclid to show that there are infinite primes.

Since there are infinite primes, there are infinite numbers divisible by primes, and we can always find a new prime p which is the only divisor of the odd number n (when p1*p1 =n).



Stephen Barry Einbinder
5/13/2013 06:17:42 am

I am somewhat fascinated by the irregularity of odd composite numbers not divisible by 3 or 5. But may I beg to differ slightly about the numbers 209 and 247?
Yes, they are both the product of two different primes.
209 is 19x11. 247 is 19x13. Might a better example be 121 as divisible only by 11, and 169 as divisible only by 13? A square of a prime number is divisible only by that prime number.

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