Timeline for Is it true that $\{x^4+y^3+z^2:\ x,y,z\in\mathbb Q_{\ge0}\}=\mathbb Q_{\ge0}$?
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14 events
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| Feb 6, 2022 at 14:14 | comment | added | Wojowu | Roth has proven that almost all positive integers are of the form $x^4+y^3+z^2$ with $x,y,z$ positive integers. | |
| Feb 5, 2022 at 9:17 | comment | added | Zhi-Wei Sun | Prof. Qing-Hu Hou has extended the verification of the 4-3-2 conjecture for $r\in\mathbb N$ with $r\le2\times10^5$. | |
| Feb 4, 2022 at 9:19 | comment | added | Zhi-Wei Sun | For $n=30000,\ldots,10^5$, there is a number $m\in\{1,2,3,4\}$ such that $m^{12}n=x^4+y^3+z^2$ for some $x,y,z\in\mathbb N$. I verified this for $30000\le n\le 40000$ and $77000\le n\le 10^5$, and Qing-Hu Hou checked this for $40000<n<77000$. | |
| Feb 4, 2022 at 6:34 | comment | added | Zhi-Wei Sun | Another related conjecture of mine states that each integer $n>1$ can be written as $x^4+y^3+z^2+2^k$ with $x,y,z\in\mathbb N$ and $k\in\{1,2,3,\ldots\}$. See oeis.org/A280356 . | |
| Feb 4, 2022 at 6:28 | comment | added | Zhi-Wei Sun | @Ilya Bogdanov In 2015 I conjectured that $\{x^4-y^3+z^2:\ x,y,z=1,2,3,\ldots\}=\mathbb Z$. See oeis.org/A266152 . | |
| Feb 3, 2022 at 23:30 | comment | added | Zhi-Wei Sun | If $m=1$ or $m=2$ is okay for $n$, then so is $m=4$. For $n=75$, $m$ cannot be $3$. For $n=1140$, $m$ cannot be $4$. For $n=23710$, $m$ must be greater than $4$. For more data, you may visit oeis.org/A350714 . For $n\le 40000$, we may require $m\le 4$ with the only exception $n=23710$ for which we may take $m=5$. | |
| Feb 3, 2022 at 22:05 | comment | added | Ilya Bogdanov | Did you try numeric studies find any nonnegative integer with no 4-3-2 integer representation? You claim you needed $m=1,2,3,4$, can this set be reduced (on the bounded segment)? | |
| Feb 3, 2022 at 6:22 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Feb 3, 2022 at 6:01 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Feb 3, 2022 at 5:40 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Feb 3, 2022 at 4:10 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Feb 3, 2022 at 3:50 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Feb 3, 2022 at 3:45 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Feb 3, 2022 at 3:38 | history | asked | Zhi-Wei Sun | CC BY-SA 4.0 |