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 2005-09-12, 02:17 #1 Orgasmic Troll Cranksta Rap Ayatollah     Jul 2003 641 Posts Prime generating polynomials that stop? Are there any polynomials that generate a finite number of primes?
 2005-09-12, 03:17 #2 jinydu     Dec 2003 Hopefully Near M48 33368 Posts p(x) = 2 Maybe you would like to clarify your question some more?
 2005-09-12, 07:51 #3 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts Challenging me to the title of Captain Obvious? Alex
 2005-09-12, 17:44 #4 Orgasmic Troll Cranksta Rap Ayatollah     Jul 2003 641 Posts 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
 2005-09-12, 18:16 #5 Phil MjX     Sep 2004 5×37 Posts 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
2005-09-12, 19:47   #6
xilman
Bamboozled!

"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across

5·17·137 Posts

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

 2005-09-12, 20:01 #7 Numbers     Jun 2005 Near Beetlegeuse 22·97 Posts Coincidentally, there is an interesting connection between polynomials that produce primes and Ulam spirals. In Paul Hoffmanns 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
2005-09-12, 22:49   #8
Orgasmic Troll
Cranksta Rap Ayatollah

Jul 2003

641 Posts

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

2005-09-12, 23:38   #9
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

2·33·139 Posts

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.

 2005-09-13, 00:26 #10 Ken_g6     Jan 2005 Caught in a sieve 5·79 Posts 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.
 2005-09-13, 00:51 #11 tom11784     Aug 2003 Upstate NY, USA 2·163 Posts 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...

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