A question about Prime Numbers

Some one sent me EMail asking me about patterns of divisors of numbers near prime numbers. The following is an edited version of my reply

A useful resource

I am only an amateur in Number Theory. I have found that the following book very helpful and a continuing pleasure to return to:

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1. From Zero to Infinity: what makes numbers interesting
2. by Constance Reid
3. (Many editions... look in libraries and 2nd hand book stores, or $7 (used) on amazon.com).

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Let me see if I have understood your question.

Question -- what can we say about numbers adjacent to prime numbers

Suppose we have a prime p, what can we say about p-1 and p+1 the numbers before and after p.

Well, if p is greater than 2 then it will not be divisible by 2. It has to be odd. There must be a remainder when we divide by 2. And this remainder has to be 1.

So

1. p = 2*n+1, for some number n=1,2,3...

   So:

2. p-1 = 2*n

   So p-1 is divisible by 2.

   More

3. p+1 = 2n+1+1 = 2*n+2 = 2*(n+1) another even number.

   So: Each prime p>2, is surrounded by two even numbers.

   Now consider 3 as a divisor. Each prime p > 3 can not be divisible by 3. There must be a (nonzero) remainder.

   This reminder is either 1 or 2.

   So, either

4. p = 3*n+1 or p=3*n+2 for some n:1.. .

   So either

5. p-1=3*n or p+1 = 3*n+3 = 3*(n+1).

   So either p-1 or p+1 is divisible by 3.

   But both p-1 and p+1 are divisible by 2.

   So either p-1 or p+1 is divisible by 6.

   MORE: consider 4: p-1 and p+1 are both even and p+1 = (p-1)+2, so one of these two numbers is divisible by 4. (the multiples of four are every other multiple of 2: 2,*4,6,*8,10,*12,...)

   SO: each prime has a multiple of 4 on one side or the other, as well as a multiple of 6.

   Is there a more general property of the intervals surrounding a prime

   Perhaps. I'm no expert. But consider a number n and a prime p that is larger than 2*n, then we know that
6. p = n*q+r for some q>1 and r:1..(n-1).

   So (visualize the numbers in line with n*q+1, n*q+2, ....n*q+(n-1) sliding below them...)

7. either n*q or n*(q+1) is within n/2 of p.

   (I guess) Example: each prime>14 is within 4 of a multiple of 7.

   That's about as far as I can go.... I think more complex patterns need some complicated Number Theory.

   I hope this helps.... don't lose the interest in numbers. They are an endless comfort and distraction... and an occasional source of fun.

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