Is this statement about primes true or false

I made a mistake in that.

It is simpler to just reuse these definitions:

z(x y) Prime pair of n definition: For a given number n and for two primes x and y, with x between 0, and y between n and 2n, if n-x is equal to y-n, then x and y are called a prime pair z(x y) of n.

Z(x y) Primal pair: For prime pair z(x y) if n-x is a prime number then prime pair z is called a primal pair Z(x y). It follows if n-x is a prime number then y-n is also a prime number as y-n = n-x

statement: There is only one n up to 8167, which has three primal pairs Z. n=18 and the three pairs are:

13,23 7,29 5,31

cheers, Jamie

For Z(count) equal to the primal pair count for a given n:

There are also no numbers n that have Z(count)=4, tested up to

8167.

There are more than one n for each Z(count) except for n=18

Coincidentally the first Z(count) which has a count shared with only a single n are n=5796 and 4380.

n Z(count)

5796 103 4380 104

This is probably a coincidence that 103 and 104 are exactly 100 bigger than Z(count) of 3 and 4 mentioned earlier, as with more n's sampled, I think more 103 and 104 Z(counts) will appear.

cheers, Jamie

Also should say that all Z(counts)=non negative integers are represented from 0 to 133, except for Z(count)=4.

The next missing Z(count) = 134, but that will probably be filled in with more n samples.

That's so difficult to understand, I'll attempt a translation. For all positive integers, x and y are a prime pair of n, p_n(x,y) when x and y are primes, 0 < x < y, n < y < 2n and n - x = y - n.

For example p_5(3,7) and p_40(41,43).

Exercise. Prove p_n(p,q) iff p,q primes, p < q, p + q = 2n.

For example, p_5(3,7) is a primal pair.

p_18(13,23) p_18(7,29) p_18(5,31)

So what? I see no significance to p_n(x,y).

Hi,

I'm not sure if your translation was correct since you missed the definition of Z(x y) which is:

for 18

18-13=prime 5 18-7=prime 11 18-5=prime 13

23-18=prime 5

29-18=prime 11 31-18=prime 13

Those are Z(x y) primal pairs for n=18

Z(13,23) Z(7,29) Z(5,31)

where the 18-13 and 23-18 gaps are the same and also equal to primes. Only number 18 has exactly 3 of them out of numbers up to 8167 (max I checked so far)

cheers, Jamie

By the way I think the statement is almost for sure false, and that if there are n that have exactly 3 of these like 18 does, they will likely be numbers that are congruent to {2, 4} mod 6

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68,...

Since numbers that like this will be a multiple of a primorial and will have greater than 3 primal pairs.

If one of these numbers exists though it will be above 8167.

cheers, Jamie

Oops meant to say:

Since numbers that AREN'T like this will be a multiple of a primorial and will have greater than 3 primal pairs.

New conjectures:

If there is a number n which is congruent to {2, 4} mod 6, and n has a Z(count) of 3, (like 18 does above) then these are true:

there are an infinite number of n that are congruent to {2, 4} mod 6 (don't have to prove that one)

there are an infinite number of n that are congruent to {2, 4} mod 6, and have a Z(count) of 1

there are an infinite number of n that are congruent to {2, 4} mod 6, and have a Z(count) of 2

there are an infinite number of n that are congruent to {2, 4} mod 6, and have a Z(count) of 3

more generally: there are an infinite number of n that are congruent to {2, 4} mod 6, and have a Z(count) of any non negative integer n

Since the Z(count) primal pairs are placed randomly from 0 to 2n, I also would conjecture that based on this:

there are an infinite number of primal pairs Z(x y) where y-x is equal to a random even number. Which shows there are infinite prime pairs etc.

cheers, Jamie

The 5,11,13 all have to be prime for those prime pairs to be counted for the definition of Z(x y).

Take for example n=20, Z(x y)=2 z(x y)=3

Proof:

0 40 1 39 2 38 3 37 *Z 20-3=17 (17=prime) 4 36 5 35 6 34 7 33 8 32 9 31 10 30 11 29 *z 20-11=9 (9 !=prime) 12 28 13 27 14 26 15 25 16 24 17 23 *Z 20-17=3 (3=prime) 18 22 19 21 20 20

So the Z(x y) count is 2, since only two of the prime pairs have a gap that is prime.

The z(x y) count is 3 though since for smaller case z prime pairs it doesn't matter if the gap is prime.

Again n=18 is the only number up to at least 8167 which has exactly three Z(x y)'s:

Z(13,23) Z(7,29) Z(5,31)

cheers, Jamie

I checked these additional ranges:

n=0 to 9945 n=15000 to 16035

there are no Z(count) of 3 or 4

My code is inefficient but I will eventually check all n up to 20000+ or more if I don't find a Z(count) of 3.

cheers, Jamie

And if:

there are no numbers n which are congruent to {2, 4} mod 6, and n has a Z(count) of 3, (like 18 does above) then that is very interesting!!!!

It would seem to imply that there is some serious order to the prime numbers like shown in this chart, where all numbers n which are congruent to {2, 4} mod 6, on the x-axis, never have a y-value greater than 2, where y-value = Z(count)

I kind of like that idea that numbers n which are: congruent to {2, 4}mod 6 never have more than two Z(x y)! Looking for a counterexample might take awhile.

cheers, Jamie