Timeline for Unexpectedly prime rich cubic polynomial
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 16, 2017 at 22:10 | comment | added | Gerry Myerson | There is a discussion of "prime-producing cubics" in a paper by Mott and Rose, available at math.fsu.edu/~aluffi/archive/paper134.ps | |
| Feb 16, 2017 at 21:43 | answer | added | Cooper Gates | timeline score: 1 | |
| Aug 14, 2015 at 12:24 | vote | accept | joro | ||
| Aug 13, 2015 at 16:32 | answer | added | Igor Rivin | timeline score: 8 | |
| Aug 13, 2015 at 12:51 | comment | added | joro | @FedorPetrov Thank you, your expectation is Bateman–Horn conjecture and is answer to Q2. I don't see how to compute the infinite product, can you? Finite product to 10^4 gives $0.0060077084$ modulo errors. | |
| Aug 13, 2015 at 11:37 | comment | added | Fedor Petrov | For $f(x)$ of degree $d$ I expect that $F(n)$ tends to $d^{-1}\prod_p (1-n_p/p)/(1-1/p)$, where $n_p$ denotes the number of roots of polynomial $f(x)$ modulo $p$. | |
| Aug 13, 2015 at 11:08 | comment | added | joro | @FedorPetrov What ratio F(n) do you expect for $f(x)=6x^2+1$ and $f(x)=6x^3+1$? Do you expect $primorial(k)x^3+1$ to give increasing ratio as $k$ gets larger? Limited numerical evidence suggests for $k=7$ the ratio is smaller at 10^n. | |
| Aug 13, 2015 at 10:56 | comment | added | Fedor Petrov | Degree 3 makes values of order $n^3$, that is, probability of being prime becomes 3 times less, where $3=\log(n^3)/\log(n)$. Small primes give another factor. Total factor may become about $1.5$, why not? | |
| Aug 13, 2015 at 10:21 | comment | added | joro | @FedorPetrov You might be right, but shouldn't the degree decrease the ratio enough for large values? | |
| Aug 13, 2015 at 10:19 | comment | added | Fedor Petrov | For polynomial, say, $f(x)=6x+1$, such ratio $F(n)$ tends to 3. I mean that the reason is likely in behaviour of $f$ modulo small primes. | |
| Aug 13, 2015 at 10:19 | history | edited | joro | CC BY-SA 3.0 |
Added data for 10^8
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| Aug 13, 2015 at 9:56 | history | asked | joro | CC BY-SA 3.0 |