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## Re: Primes and Squares1 - Clarifications2

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• The best way to avoid torturing a mathematician is to say nothing, or maybe talk about the weather. Barring that, clarity and conciseness expressed in terms
Message 1 of 2 , Jan 2, 2004
The best way to avoid torturing a mathematician is to say nothing, or
maybe talk about the weather. Barring that, clarity and conciseness
expressed in terms familiar to them is the only medicine that eases
the pain. Remember, short does not guarantee concise. Even us
amateur geniuses aren't smart enough to always know (or even
discover) the correct mathematical term to use in a given instance.
But, even though we might not have the correct terminology at our
disposal, making up new terminology only makes matters worse and
should be avoided if possible. The statement "odd divisors not
exceeding n" would be understood, whereas a new term "primary
divisors" requires a definition and increases the
reader's "conceptual burden" <--Do you know what that term means?,
neither do I.

The mirror sequence you speak of is well known and plays a role in
the foundations of many theorems and allows prime hunters to save
tons of computer cycles, which must be a lot since they don't have
mass.

Suggested reading - Wilson's Theorem, Quadratic Reciprocity,
LeGendre's Prime Counting function, the theories of linear and

Keywords - primorial, factorial, sigma divisor function, Euler
Totient function, Jacobi symbol...

...anybody else have suggestions?

> MIRROR SEQUENCE clarification
> As I stated in my original submittal, the mirror sequence is a
> sequence of primary odd divisors that is centered at 2*the product
of
> all odd integers to N and spans +/- N from that center.

The divisor structure you describe centered on ...20789, 20791... ,
is exactly the same as that of ...-1, 1... , there is no need to look
at such a large number only for this information. That number, by
the way, = n!! for odd n, where !! is the double factorial function.

The density distribution of all odd primes <= sqrt(n) as relates to
their occurence as a factor in a sequence of n consecutive odd
integers, is neither maximum nor a minimum when centered over {-
n,...,n} and cannot be used as a bound.

Here's an example
For, {1,2,3,4,5,6,7,8,9}, 3 divides elements {3,6,9}, 3 out 9
elements, density=1/3
For, {1,3,5,7,9,11,13,15}, same result, 3 divides 3 out of 9 elements
For, {15,17,19,21,23,25,27}, 3 divides 3 out of 7 elements > 1/3
For, {1,3,5,7,9,11,13,15,17,19}, 3 divides 3 out of 10 elements < 1/3

The pattern within n odd numbers centered on n!! is the exact same
pattern as the n odd numbers centered on zero. All such patterns for
any given span of n consecutive odd numbers, transposed say, a
distance x from the center point at 0, will also occur as a mirror
image at n!!-x, this is true. In fact, all unique, possible
combinations are exhausted as the center point moves from 0 to 2 to
4...to (n!!-2) to n!! Despite being "equivalent", the mirror image
pairs are unique from each other within our frame of reference of
increasing odd integers. Except that the sequences centered on 0 and
n!! are not only equivalent, the are equal and identical. The
sequence centered on 3 is identical to that centered on n!! + 3,
while it's mirror image occurs at n!!-3 and repeats every k*n!!
transposition of it's center, k = -inf, +inf. Thus there are n!!
unique divisor distribution patterns for a sequence of n consecutive
odd integers when only the odd divisors not exceeding n are
considered in the make-up of the pattern. If mirror images are not
considered unique, there are (n!!+1)/2 unique distributions.

> Now lets match up the REF set with it's divisors from the LE
> set.
>
> REF set
> 20779 20781 20783 20785 20787 20789 20791 20793 20795
20797
> 20799 20801
>
> LE set values
> 11 9 7 5 3 X X 3 5
7
> 9 11
>
> Of course, there are odd divisors higher than 11
> that may divide 20789 and/or 20791, but let's not bother to check,
> since it is not germane to this argument

Au contraire.. there must be and there are odd divisors higher than
11 that do divide those numbers. The same is true for most all odd
numbers greater than 11. And one must bother to check them, or at
least account for them, before a conclusion can be drawn.

In a nutshell, I think you are focusing on numbers relatively prime
to n as if all of them were primes, but only a subset of the integers
relatively prime to n are actually primes. It almost looks that way
for small n, but in reality, the ratio of primes < n over integers
less than and relatively prime to n, tends to zero with increasing n.

-Dick
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