Author |
Topic: Polynomial and Prime Numbers (Read 1823 times) |
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender: 
Posts: 13730
|
 |
Re: Polynomial and Prime Numbers
« Reply #25 on: Sep 19th, 2003, 1:03am » |
 Quote  Modify
|
on Sep 18th, 2003, 11:18pm, Barukh wrote:| That's very interesting! I would like to get a proof... |
|
The proof is allready spread around in the thread.
TenaliRaman proved that for any prime p, and natural numbers m and q
P(m+pq) = p + k*p with some integer k
And I then proved that if P is finite k can't be 0 for all m and q, and thus P(m+pq) will at some point be a multiple of p (which by definition isn't a prime).
So, even without the restriction of integer coefficients, finite order polygons can't be prime for every n, but some infinite polygons can. And thanks to James, we allready have an example of such an infinite polynomial.
|
|
 IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
|
Re: Polynomial and Prime Numbers
« Reply #26 on: Sep 19th, 2003, 6:38am » |
Quote Modify
|
on Sep 19th, 2003, 1:03am, towr wrote:
So, even without the restriction of integer coefficients, finite order polygons can't be prime for every n, but some infinite polygons can. And thanks to James, we allready have an example of such an infinite polynomial. |
|
What I had in mind, is the proof that there exists/doesn't exist a polynomial of infinite degree with integer coefficients satsifying the conditions of the problem. The polynomial in James's example isn't with integer coefficients.
|
|
IP Logged |
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: Polynomial and Prime Numbers
« Reply #27 on: Sep 19th, 2003, 4:00pm » |
Quote Modify
|
To be convergent for all natural numbers, a power series (sorry - I don't care for calling them "infinite polynomials") must have coefficients that converge to zero. The only way this can happen and still have all integer coefficients is for the coefficients to be eventually zero. I.e. for it to be a "finite" polynomial after all!
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
TenaliRaman
Uberpuzzler
I am no special. I am only passionately curious.
Gender:
Posts: 1001
|
|
Re: Polynomial and Prime Numbers
« Reply #28 on: Sep 20th, 2003, 4:39am » |
Quote Modify
|
(Sorry for Replying Late!)
Thx Sir Col!!
i agree its the best way to learn but its no fun!(and i should know because when i get some answer right in these forum, its like i have achieved something really big and compare it with the times i get my answers wrong, the ratio is just around zero)
@James,
umm now why din't i think of that ?? 
|
|
IP Logged |
Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Polynomial and Prime Numbers
« Reply #29 on: Sep 20th, 2003, 4:42am » |
Quote Modify
|
on Sep 19th, 2003, 6:38am, Barukh wrote:
What I had in mind, is the proof that there exists/doesn't exist a polynomial of infinite degree with integer coefficients satsifying the conditions of the problem. The polynomial in James's example isn't with integer coefficients. |
|
Yes, I know, but I was content to first prove it had to be infinite if it could at all satisfy the criteria of being prime for all n.
on Sep 19th, 2003, 4:00pm, Icarus wrote:| To be convergent for all natural numbers, a power series (sorry - I don't care for calling them "infinite polynomials") must have coefficients that converge to zero. The only way this can happen and still have all integer coefficients is for the coefficients to be eventually zero. I.e. for it to be a "finite" polynomial after all! |
|
Does it need to be convergent? It seems to me the set of primes converges, as doers the set of natural numbers..
Or, more likely, do I simpyl not grasp what you mean with convergent in this context?
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: Polynomial and Prime Numbers
« Reply #30 on: Sep 20th, 2003, 5:26am » |
Quote Modify
|
I think he meant that for any power series to be convergent:
P(x)=a0+a1x+a2x2+... (to infinity)
It is a necessary condition that as k[to][infty], ak[to]0.
Otherwise, its value, for any n, would be undefined. E.g. P(x)=1–2x+3x2–4x3+..., what would P(3) be?
Hence it cannot both be a convergent infinite series (for natural numbers) and contain integer coefficients in every term.
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
|
Re: Polynomial and Prime Numbers
« Reply #31 on: Sep 20th, 2003, 7:05am » |
Quote Modify
|
on Sep 19th, 2003, 4:00pm, Icarus wrote:| To be convergent for all natural numbers, a power series (sorry - I don't care for calling them "infinite polynomials") must have coefficients that converge to zero. The only way this can happen and still have all integer coefficients is for the coefficients to be eventually zero. I.e. for it to be a "finite" polynomial after all! |
|
Beautiful argument, Icarus! And such simple...
|
|
IP Logged |
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: Polynomial and Prime Numbers
« Reply #32 on: Sep 20th, 2003, 5:50pm » |
Quote Modify
|
on Sep 20th, 2003, 4:42am, towr wrote:Does it need to be convergent? It seems to me the set of primes converges, as doers the set of natural numbers..
Or, more likely, do I simpyl not grasp what you mean with convergent in this context? |
|
Actually I do not grasp what you could possibly mean by "convergent" in those sentences!
The set of primes converges?  The Natural Numbers converge? To what? Infinity would work, but since we are stuck in the Real numbers here, that is called "divergence".
What I meant by convergence is: The sequence of partial sums has a finite limit. Does an infinite power series need to converge? Yes, if it is going to take on any values. Since this one is required to have values for every natural number, it's radius of convergence must be [infty].
on Sep 20th, 2003, 7:05am, Barukh wrote:| Beautiful argument, Icarus! And such simple... |
|
What can I say? "Simple" is my specialty. The tough ones somebody else has to handle! 
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Polynomial and Prime Numbers
« Reply #33 on: Sep 21st, 2003, 8:49am » |
Quote Modify
|
on Sep 20th, 2003, 5:50pm, Icarus wrote:| Actually I do not grasp what you could possibly mean by "convergent" in those sentences! |
|
What I meant with converge was diverge, obviously  (doesn't everyone?)
Quote:| What I meant by convergence is: The sequence of partial sums has a finite limit. Does an infinite power series need to converge? Yes, if it is going to take on any values. Since this one is required to have values for every natural number, it's radius of convergence must be [infty]. |
|
Makes sense now..
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
ecoist
Senior Riddler
Gender:
Posts: 405
|
|
Re: Polynomial and Prime Numbers
« Reply #34 on: Mar 1st, 2006, 7:59pm » |
Quote Modify
|
Did anyone notice that TenaliRaman's proof suggests that polynomials are very bad prime generators?
Given any nonconstant polynomial p(x) with integer coefficients and any positive integer n, there exists an integer x_n such that p(x_n) is divisible by at least n distinct primes.
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print