Timeline for Diophantine equations with arithmetical functions
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| May 18, 2022 at 1:17 | comment | added | Gerry Myerson | Because they are there. | |
| May 17, 2022 at 7:25 | comment | added | user482376 | Thank you so much. But why do they need to solve these type of equations? | |
| May 17, 2022 at 7:20 | comment | added | Gerry Myerson | B42, behavior of $\phi(\sigma(n))$ and $\sigma(\phi(n))$. And more. | |
| May 17, 2022 at 7:18 | comment | added | Gerry Myerson | Here are some questions discussed in Guy, Unsolved Problems In Number Theory, 3rd edition. B11, solutions of $m\sigma(m)=n\sigma(n)$. $\sigma(a)/a=\sigma(b)/b$. $\sigma(n)=({\rm rad\ } n)^2$, where rad is the radical of $n$. B12, analogues with $d(n)$, $\sigma_k(n)$. B13, solutions of $\sigma(n)=\sigma(n+1)$ (and variations). B15, solutions of $\sigma(q)+\sigma(r)=\sigma(q+r)$. B18, solutions of $d(n)=d(n+1)$. B36 is devoted to $\phi(n)$. B37, does $\phi(n)$ divide $n-1$ for some composite $n$? B38, solutions of $\phi(m)=\sigma(n)$. B41, iterations of $\phi$ and $\sigma$. (continued) | |
| May 17, 2022 at 6:58 | comment | added | Gerry Myerson | $\sigma(x)-\phi(x)\ge(x+1)-(x-1)=2$ for $x\ge2$. | |
| May 17, 2022 at 6:32 | comment | added | user482376 | Yes, I have tried to explain more about it there. | |
| May 17, 2022 at 6:26 | comment | added | Martin Sleziak | This seems to be quite close to your other question: Diophantine equations or associative operations on ordered lattice. | |
| May 17, 2022 at 6:25 | comment | added | user482376 | $\sigma(x)$ is the sum of divisors of $x$. | |
| May 17, 2022 at 6:24 | history | edited | user482376 | CC BY-SA 4.0 |
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| May 17, 2022 at 5:54 | comment | added | Fedor Petrov | your examples, especially the second one, look strange | |
| May 17, 2022 at 5:53 | history | edited | Fedor Petrov | CC BY-SA 4.0 |
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| May 16, 2022 at 22:27 | answer | added | Joe Silverman | timeline score: 2 | |
| May 16, 2022 at 21:35 | history | edited | user482376 | CC BY-SA 4.0 |
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| May 16, 2022 at 21:33 | comment | added | user482376 | Is this topic not good as topic of research ? | |
| May 16, 2022 at 21:13 | history | edited | user482376 | CC BY-SA 4.0 |
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| May 16, 2022 at 20:50 | comment | added | Joe Silverman | These are not typically called Diophantine equations, which generally refer to systems of pol\iynomial equations. But equations of the sort you're asking abouot are often studied. For example, the equation $\sigma(n)=2n$ characterizes perfect numbers. And there are also equations involving the Euler $\phi$ function that have been much studied. | |
| May 16, 2022 at 20:20 | history | asked | user482376 | CC BY-SA 4.0 |