Prime generating polynomials that stop?

[Orgasmic Troll]

Are there any polynomials that generate a finite number of primes?

[jinydu]

p(x) = 2

Maybe you would like to clarify your question some more?

[akruppa]

Challenging me to the title of Captain Obvious?

Alex

[Orgasmic Troll]

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

[Phil MjX]

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.

[xilman]

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

[Numbers]

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

[Orgasmic Troll]

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

[R.D. Silverman]

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.

[Ken_g6]

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!

[tom11784]

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...