Timeline for Prime generating polynomials
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 13, 2018 at 22:20 | comment | added | Delmastro | @GregMartin What a pity! Oh well, I guess I was being overly optimistic. Thank you for your comment though, I appreciate the input. | |
| Nov 13, 2018 at 22:18 | comment | added | Delmastro | @StanleyYaoXiao Oh, I now I see what you meant. Kind of anticlimactic, but I guess that's the way it goes. Anyway, thank you very much for your insight! | |
| Nov 13, 2018 at 18:26 | comment | added | Greg Martin | To say the same thing another way: the strategies that number theorists have come up with for trying to show that a given polynomial must represent at least one prime (other than brute-force computation) are all actually strategies for showing that it must represent infinitely many primes. It seems hard to think of a theoretical approach to proving "one prime" that doesn't also prove "infinitely many primes". | |
| Nov 13, 2018 at 17:54 | comment | added | Stanley Yao Xiao | $x^2 + 1$ obviously represents 5, because you can find an explicit $x$ such that $x^2 + 1$ is equal to 5. This is not doable for a generic quadratic polynomial, say $ax^2 + bx + c$. So outside of obvious 'eyeballing', one can't prove that a quadratic polynomial represents a prime at all. If one can in fact show that a generic quadratic polynomial represents a prime, then very likely the argument will in fact produce infinitely many primes that it can represent. | |
| Nov 13, 2018 at 17:48 | comment | added | Delmastro | @StanleyYaoXiao Thank you for your comment, but I must obviously missing something. "It represents a prime" cannot be equivalent to "it represents infinitely many primes". For one thing, $x^2+1$ clearly represents $2^2+1=5$, one prime, but it is not known if it represents infinitely many. So "representing one prime" is a much weaker statement than "it represents infinitely many". Or did I misunderstand your comment? | |
| Nov 13, 2018 at 16:49 | comment | added | Stanley Yao Xiao | No. Typically we expect that if we can show a polynomial that doesn't obviously represent a prime represents a prime at all then it represents infinitely many primes. At least, that's a corollary of all successful proofs so far. | |
| Nov 13, 2018 at 16:45 | history | asked | Delmastro | CC BY-SA 4.0 |