Arithmetic and Polynomial Progression of Primes?

[drake2]

Interesting paper http://arxiv.org/PS_cache/math/pdf/0610/0610050.pdf .

I barely understand any of it.

Can someone give me an example of primes in polynomial progression?

Is there also a longest known set of primes in polynomial progression like there are longest known sets of primes in arithmetic progression?

[R.D. Silverman]

Originally Posted by drake2

Interesting paper http://arxiv.org/PS_cache/math/pdf/0610/0610050.pdf .

Can someone give me an example of primes in polynomial progression?

Is there also a longest known set of primes in polynomial progression like there are longest known sets of primes in arithmetic progression?

Let P1(z) = z, P2(z) = z^2 + z, P3(z) = z^3 + z

Then (3,7,11) = (1 + P1(2), 1 + P2(2), 1 + P3(2)) (x = 1)

The fact that this is also an AP is a coincidence because of the polynomials
that I chose and that I evaluated it at 2.

Evaluate it at z = 4 and you get the polynomial progression 7,23,71 with
(x = 3)

[grandpascorpion]

I noticed the next two terms in the sequence are primes as well (263 and 1071)

One small thing: In order to get more than two terms in your sequence (unless z=2 and x=1), the following must be true: z%6 = 4 and x%6 = 3

[grandpascorpion]

z=4, x=33 yields a sequence of 7 primes: 37, 53, 101, 293, 1061, 4133, 16421

Can anyone find a longer one?

[drake2]

I think I may have learned something.

How about z=4 x=159 to get:
163, 179, 227, 419, 1187, 4259, 16547, 65699

[drake2]

This seems to be the longest polynomial progression of primes for z=4 and x<100000000 for the particular polynomials.

z=4
x=33890799

33890803, 33890819, 33890867, 33891059, 33891827, 33894899, 33907187, 33956339, 34152947, 34939379, 38085107

[drake2]

Sequence of 12 primes in polynomial progression

z=16
x=244866027

244866043, 244866299, 244870139, 244931579, 245914619, 261643259, 513301499, 4539833339, 68964342779, 1099756493819, 17592430910459, 281475221576699

Can anyone find a sequence of 13?

[grandpascorpion]

z=16
x=836880681

836880697,836880953,836884793,836946233,837929273,
853657913,1105316153,5131847993,69556357433,1100348508473,
17593022925113,281475813591353,4503600464251193

[R.D. Silverman]

Originally Posted by grandpascorpion

z=16
x=836880681

836880697,836880953,836884793,836946233,837929273,
853657913,1105316153,5131847993,69556357433,1100348508473,
17593022925113,281475813591353,4503600464251193

The Tao paper shows that there exists arbitarily long progressions.

So what point is there to these posts? They are not even numerical curiosities.

I can easily find such progressions of length 50, 100, 1000, etc.

Given *any* set of increasing primes, I can find a polynomial to
match. These posts are *pointless*.


I can find a polynomial and a value of x so that

x + (f(1), f(2), f(3), f(4), ....) = 3,5,7,11,13,17,19,23,29,31,....

Or for ANY specified set of primes on the right.

Finding a specific instance for one value of x is *meaningless*.

Try finding a polynomial and TWO different values of x etc. such that
x1 + f(1), x1 + f(2), etc.

AND x2 + f(1), x2+ f(2), x2 + f(3) etc. are prime. Or find 3 such x.

The Tao result is fascinating. These numerical examples are a total waste
of computer and people time.

[Jens K Andersen]

Originally Posted by R.D. Silverman

Try finding a polynomial and TWO different values of x etc. such that
x1 + f(1), x1 + f(2), etc.

AND x2 + f(1), x2+ f(2), x2 + f(3) etc. are prime. Or find 3 such x.

The Tao result is fascinating. These numerical examples are a total waste
of computer and people time.

There are few Tao's in the world. Some people just like a computational challenge, even when it doesn't lead to results which are publishable or of wide interest. I think there should be room for that here.
Choosing the polynomials first like your z, z^2+z, z^3+z, ... makes it a challenge.

Your suggestion to find 2 or 3 values of x with presumably arbitrary polynomials is no challenge.
Just compute e.g. the first million prime triplets on form (p, p+2, p+6). Then choose a polynomial of degree one million to run through the p values, and let x1=0, x2=2, x3=6.
The solution would not be suited for listing here (or anywhere else).

Restrictions are needed to make it computationally interesting. Here is a variant with two x values leading to many consecutive primes:
If x is 46863043340173267 or 534402442999154537, then
x + 0,2,56,62,80,110,146,150,152,170,230,234,252,264,276,290,294,296,300,302,344 are 21 consecutive primes. Can anybody beat that?
Computing a corresponding polynomial would be a simple exercise.

[grandpascorpion]

Originally Posted by R.D. Silverman

So what point is there to these posts? They are not even numerical curiosities.

I can easily find such progressions of length 50, 100, 1000, etc.

Given *any* set of increasing primes, I can find a polynomial to
match. These posts are *pointless*.

That's fine. Of course, you can fit a single polynomial to any set of points. I just thought there was something significant in the progression of your example which is an exponential function (except for the first term): f(n)=z^n+z+x.

Oh well, that is what I get for not looking at the paper.