Timeline for gcd of three numbers
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10 events
| when toggle format | what | by | license | comment | |
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| Jan 1, 2012 at 0:46 | vote | accept | user3208 | ||
| Dec 31, 2011 at 18:10 | history | edited | user631 | CC BY-SA 3.0 |
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| Dec 31, 2011 at 10:04 | comment | added | GH from MO | I see Gerhard Paseman made a similiar comment. In fact one can replace $n$ by its squarefree part in the original problem. Also, I believe Iwaniec's estimate is effective, hence the implied constant in $c,d\ll\log^2 n$ can be calculated (in principle). | |
| Dec 31, 2011 at 9:51 | comment | added | GH from MO | Some comments: 1. The assumption $(a,b)=1$ can be omitted as it is not used in the proof. 2. Iwaniec even proved $j(n)\ll r^2\log^2 r$, where $r$ is the number of distinct prime factors of $n$, see Demonstratio Math. 11 (1978), 225-231 (MR0499895). As $r\ll\log n/\log\log n$, we have a solution $c,d\ll\log^2 n$ in the OP's question. | |
| Dec 31, 2011 at 9:23 | comment | added | Alan Haynes | @The Hamburglar: Aha, I wasn't aware of Iwaniec's result- can you give a reference? | |
| Dec 31, 2011 at 8:30 | history | edited | user631 | CC BY-SA 3.0 |
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| Dec 31, 2011 at 7:49 | history | edited | user631 | CC BY-SA 3.0 |
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| Dec 31, 2011 at 7:28 | comment | added | Gerhard Paseman | Some minor tweaks: j(n) depends only on the distinct prime divisors of n, so one can replace n in the upper bound by the squarefree part of n. Further, letting r < log(n) be the number of distinct prime divisors, Iwaniec's result actually implies j(n) << (rlog(r))^2. Finally, mathoverflow.net/questions/37679/… and another answer to the same question give alternative explicit bounds on j(n) by Kanold, Stevens, and Paseman with small nonconstant exponent. Gerhard "Ask Me About Jacobsthal's Function" Paseman, 2011.12.30 | |
| Dec 31, 2011 at 6:47 | history | edited | user631 | CC BY-SA 3.0 |
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| Dec 31, 2011 at 6:39 | history | answered | user631 | CC BY-SA 3.0 |