Timeline for How to recover integer part from known fractional root part?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 13 at 19:40 | answer | added | Steven Stadnicki | timeline score: 5 | |
| Jan 13 at 4:03 | answer | added | Daniel Weber | timeline score: 2 | |
| Jan 12 at 22:50 | comment | added | Steven Stadnicki | It's not too hard to show that you can only have one value of $n$ that makes $r^2$ an (exact) integer, incidentally; if $r=n+f$ with $r^2=q$, say, then $\left((m+n)+f\right)^2$ $= \left(m+(n+f)\right)^2$ $= m^2+(n+f)^2+2m(n+f)$ $=m^2+q+2m(n+f)$ and clearly the last term is irrational, since $f$ is and $m, n, q$ are all integers. But I'm not even sure if good lower bounds are known for $\min_{m,n\leq N}\left(\left|\sqrt{n}-\sqrt{m}\right|\bmod 1\right)$ as a function of $N$. | |
| Jan 12 at 22:34 | history | edited | ReverseFlowControl | CC BY-SA 4.0 |
Improve question based on comment.
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| Jan 12 at 22:27 | comment | added | ReverseFlowControl | @StevenStadnicki, thank you! I will add an edit to the question. | |
| Jan 12 at 22:17 | answer | added | Marco Ripà | timeline score: 0 | |
| Jan 12 at 22:12 | comment | added | Steven Stadnicki | Without a bound on $r$, the answer is no; square roots are dense mod 1, so however close an approximation to $f$ you have, there are infinitely many square roots of whole numbers whose fractional part is that close to $f$. | |
| Jan 12 at 21:16 | history | asked | ReverseFlowControl | CC BY-SA 4.0 |