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

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

So:
	p-1 = 2*n

So p-1 is divisible by 2.

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

So either
	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
	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...)
	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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