mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math

Reply
 
Thread Tools
Old 2005-09-12, 02:17   #1
Orgasmic Troll
Cranksta Rap Ayatollah
 
Orgasmic Troll's Avatar
 
Jul 2003

28116 Posts
Default Prime generating polynomials that stop?

Are there any polynomials that generate a finite number of primes?
Orgasmic Troll is offline   Reply With Quote
Old 2005-09-12, 03:17   #2
jinydu
 
jinydu's Avatar
 
Dec 2003
Hopefully Near M48

175810 Posts
Default

p(x) = 2

Maybe you would like to clarify your question some more?
jinydu is offline   Reply With Quote
Old 2005-09-12, 07:51   #3
akruppa
 
akruppa's Avatar
 
"Nancy"
Aug 2002
Alexandria

2,467 Posts
Default

Challenging me to the title of Captain Obvious?

Alex
akruppa is offline   Reply With Quote
Old 2005-09-12, 17:44   #4
Orgasmic Troll
Cranksta Rap Ayatollah
 
Orgasmic Troll's Avatar
 
Jul 2003

641 Posts
Default

Okay, egg on my face ;)

I was thinking about n2+n+41, and as far as I know, it is unknown whether it produces an infinite number of primes.

I was thinking that maybe there would be some value excluding trivial solutions, but I'm not entirely sure what would constitute "trivial" and think I just asked a bonehead question
Orgasmic Troll is offline   Reply With Quote
Old 2005-09-12, 18:16   #5
Phil MjX
 
Phil MjX's Avatar
 
Sep 2004

5×37 Posts
Default

Hi !

Just a related question (but the converse of TravisT absence of primes): the Ulam spiral produces diagonals rich in primes corresponding to 2nd polynomials : an^2+bn+c...In a book called "merveilleux nombres premiers" by JP Delahaye, it is written that this isn't well understood...can you give me some links or explanations about it?

Thanks.
Philippe.

Last fiddled with by Phil MjX on 2005-09-12 at 18:17
Phil MjX is offline   Reply With Quote
Old 2005-09-12, 19:47   #6
xilman
Bamboozled!
 
xilman's Avatar
 
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

101001011001102 Posts
Default

Quote:
Originally Posted by TravisT
...but I'm not entirely sure what would constitute "trivial" and think I just asked a bonehead question
Got it in one!

Give "trivial" a non-trivial definition and the question is probably not bone-headed.

Here is a quadratic that generates only one prime number when evaluated at integer arguments: p(x) = 2x^2. Note that, unlike the earlier example, this one generates an infinite number of different values but only one is prime.

Paul
xilman is offline   Reply With Quote
Old 2005-09-12, 20:01   #7
Numbers
 
Numbers's Avatar
 
Jun 2005
Near Beetlegeuse

6048 Posts
Default

Coincidentally, there is an interesting connection between polynomials that produce primes and Ulam spirals. In Paul Hoffmanns book The Man Who Loved Only Numbers a biography of Paul Erdos there is an explanation of how Stan Ulam came to discover these spirals. He was doodling while listening to a long boring paper at a math conference, and wrote the number 1 in the middle then wrote successive integers in a counter-clockwise square spiral.

17 16 15 14 13
18--5--4-3--12
19--6--1--2-11
20--7--8--9-10

and so on. He noticed that the primes fall on common diagonals. After playing with this idea he found that if you start in the middle with 17 instead of 1 and do the same thing, you end up with a long diagonal containing the primes 227, 173, 127, 89, 59, 37, 23, 17, 19, 29, 47, 73, 107, 149, 199, 257. And the connection with polynomials? Well, the polynomial n^2 + n + 17, discovered by Euler, produces these exact same primes. Curious, but true.

I should add that this only works for n = 0 to 15

Last fiddled with by Numbers on 2005-09-12 at 20:06 Reason: Added range for n
Numbers is offline   Reply With Quote
Old 2005-09-12, 22:49   #8
Orgasmic Troll
Cranksta Rap Ayatollah
 
Orgasmic Troll's Avatar
 
Jul 2003

641 Posts
Default

Quote:
Originally Posted by xilman
Got it in one!

Give "trivial" a non-trivial definition and the question is probably not bone-headed.

Here is a quadratic that generates only one prime number when evaluated at integer arguments: p(x) = 2x^2. Note that, unlike the earlier example, this one generates an infinite number of different values but only one is prime.

Paul
After some very brief noodling, I'm going to tentatively define a trivial solution as a polynomial that produces exactly 1 prime
Orgasmic Troll is offline   Reply With Quote
Old 2005-09-12, 23:38   #9
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

22·5·373 Posts
Default

Quote:
Originally Posted by TravisT
After some very brief noodling, I'm going to tentatively define a trivial solution as a polynomial that produces exactly 1 prime
How about a polynomial that produces NO primes? There are, of course
infinitely many examples of this type.

f(x) = (x-1) (x-6) or f(x) = (x-3)(x-2)(x-5)(x-7) etc.
R.D. Silverman is offline   Reply With Quote
Old 2005-09-13, 00:26   #10
Ken_g6
 
Ken_g6's Avatar
 
Jan 2005
Caught in a sieve

5·79 Posts
Default

I'm assuming you're looking for f(x), where for x any nonnegative integer, there are only N>1 cases where f(x) is prime?

Okay, try this on for size :

f(x) = 3-x

f(0) = 3, Prime!
f(1) = 2, Prime!
f(2) = 1, not prime!
f(3) = 0, not prime!
for x>=4, f(x) < 0; therefore f(x) is not prime!

Last fiddled with by Ken_g6 on 2005-09-13 at 00:28 Reason: "whole number" is not well defined.
Ken_g6 is offline   Reply With Quote
Old 2005-09-13, 00:51   #11
tom11784
 
tom11784's Avatar
 
Aug 2003
Upstate NY, USA

2×163 Posts
Default

f(x) = n^2+n+41 - X*floor( (n^2+n+41)/X )
where X is some bound that we do not want our function to produce values above
probably not exactly what you're looking for though...
tom11784 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Error generating or reading NFS polynomials Brownfox Msieve 7 2018-04-06 16:24
Prime generating series Citrix Open Projects 18 2013-08-24 05:24
when does prime seach stop? Unregistered Information & Answers 5 2011-08-10 01:38
LLR 3.8.2: more flexible stop-on-prime option mdettweiler Conjectures 'R Us 21 2010-10-03 13:38
Prime-generating polynomials Orgasmic Troll Math 28 2004-12-02 16:37

All times are UTC. The time now is 20:12.

Mon Mar 8 20:12:16 UTC 2021 up 95 days, 16:23, 1 user, load averages: 0.57, 1.10, 1.34

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.