Timeline for Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 19, 2021 at 19:56 | comment | added | Maciej Ulas | @ASP I think that the best idea is to check the papers cited in Badziahin's paper and the papers which cite it. | |
| Oct 19, 2021 at 16:28 | comment | added | ASP | @Maciej Ulas, the paper you cited by D. Badziahin has been really very useful in my research. The author has classified all divisors of $P(x) =x^4 +c_1x^3 +c_2x^2 +c_3x+1$ that are congruent to $1$ modulo $x$. Before I attempt to extend the ideas of this paper with my supervisor, are there any other papers building on this research or similar papers that classify divisors of numbers $P(n) $ where $P$ is a polynomial of degree greater than $4$? | |
| Sep 19, 2021 at 0:27 | answer | added | Samuel | timeline score: 0 | |
| Sep 13, 2021 at 22:11 | answer | added | Jamie | timeline score: 5 | |
| Sep 9, 2021 at 17:44 | comment | added | ASP | arxiv.org/abs/2005.02327 (General purpose primality test, Theorem 2.2) | |
| Sep 9, 2021 at 17:35 | comment | added | ASP | The paper is great. It considers a more general polynomial $P(z) = z ^4 +c_1z^3 +c_2z^2 +c_3z+1$. Just wondering, are there any methods that can decide whether $P(z) $ has a proper divisor $d \equiv 1 $ (mod $z) $ where $P(z) $ is a polynomial of degree $n>4$. If such methods exist, then a faster primality test exists for numbers $P(z) $ (faster than the current $N-1 $ tests). | |
| Sep 9, 2021 at 14:52 | comment | added | Maciej Ulas | You should consult the paper of D. Badziahin (Finding special factors of values of polynomials at integer points, Int. J. of Numb. Theor., V. 13(1), 2017), where slightly more general problem was considered, i.e., the existence, for a given polynomial $P$, divisors $d$ of $P(n)$ such that $d\equiv 1\pmod{n}$. | |
| Sep 9, 2021 at 10:57 | comment | added | JoshuaZ | Regarding your last question: if $n+1$ is not prime then $z^n + z^{n-1}... +1$ will not be irreducible(and will in fact factor into non-trivial cyclotomic polynomials), so the behavior of general solutions to this will likely be not great. Note also that there's a preprint on Arxiv by Sean Bibby, Pieter Vyncke and myself which looks at a related sort of Diophantine equation arxiv.org/abs/1908.09420 . What we call a $\sigma_{4,1}$ quasisolution there is similar to what you are looking at but with the left hand sides +1s replaced with -1s. Similar techniques may be of use/interest. | |
| Sep 9, 2021 at 10:51 | history | edited | ASP | CC BY-SA 4.0 |
fixed a typo
|
| Sep 9, 2021 at 10:31 | review | Close votes | |||
| Sep 12, 2021 at 19:47 | |||||
| Sep 9, 2021 at 10:08 | history | edited | ASP | CC BY-SA 4.0 |
deleted 79 characters in body
|
| Sep 9, 2021 at 10:03 | history | edited | ASP | CC BY-SA 4.0 |
deleted 79 characters in body
|
| Sep 9, 2021 at 9:50 | history | asked | ASP | CC BY-SA 4.0 |