Timeline for Irreducibility of the polynomial $x^n+5x+3$ over $\mathbb{Q}$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 22 at 5:07 | vote | accept | Clario | ||
| Mar 22 at 1:18 | vote | accept | Clario | ||
| Mar 22 at 5:07 | |||||
| Mar 21 at 20:36 | history | edited | Will Sawin | CC BY-SA 4.0 |
added 62 characters in body
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| Mar 21 at 20:24 | comment | added | GH from MO | This is fun stuff, I did not know about these works and techniques. In particular, I did not know that Erdős covering systems were relevant here. Too bad I am too busy to study this theory. I have already learned a bit, thanks to you. | |
| Mar 21 at 20:20 | comment | added | Will Sawin | @GHfromMO Thanks for pointing that out! Combining that and the calculations already in my answer, I can do a bit better, will do soon. | |
| Mar 21 at 19:51 | comment | added | GH from MO | The paper Filaseta-Li-Patane-Skabelund: On the irreducibility of the non-reciprocal part of polynomials of the form $f(x)x^n+g(x)$, Acta Arithmetica 196 (2020), 187-201, builds on your paper. Their Corollary 1 gives that $x^n+5x+3$ is irreducible for $n>24157816$. | |
| Mar 21 at 19:36 | comment | added | GH from MO | Very nice. I looked up Filaseta-Ford-Konyagin: On an irreducibility theorem of A. Schinzel associated with coverings of the integers (2000), and their Corollary on p.635 tells us that $x^n+5x+3$ is irreducible for $n\geq 2\cdot 5^{137}$. An unspecified but effectively computable sufficient lower bound for $n$ already follows from Theorem 3 in Schinzel: On the reducibility of polynomials and in particular of trinomials (1965). | |
| Mar 21 at 13:16 | history | edited | YCor | CC BY-SA 4.0 |
fixed norm signs (|| -> \|)
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| Mar 21 at 13:05 | history | answered | Will Sawin | CC BY-SA 4.0 |