Timeline for Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 23, 2010 at 17:20 | answer | added | Max Alekseyev | timeline score: 2 | |
| Jul 23, 2010 at 13:54 | vote | accept | Ken Fan | ||
| Jul 23, 2010 at 6:37 | comment | added | Gerry Myerson | Numbers of the form $a^2b^3$ (without the restriction $a>1$, $b>1$) are known as powerful numbers, or squarefull numbers. Knowing this vocabulary may help you find something in the literature. It's known that every integer can be expressed as a difference of powerful numbers in infinitely many ways. It's unknown whether there are three consecutive powerful numbers. (The two preceding sentences are not directly related to your question, I just thought I'd throw them in.) | |
| Jul 23, 2010 at 5:09 | answer | added | user631 | timeline score: 25 | |
| Jul 23, 2010 at 5:00 | comment | added | Wadim Zudilin | BTW, even in the case of "small" $a^2-b^3$ for $a,b\in\mathbb Z$ the results on the finiteness seem to be nontrivial, see Marshall Hall's conjecture (en.wikipedia.org/wiki/Hall%27s_conjecture), especially the link to the page of Noam Elkies. | |
| Jul 23, 2010 at 4:52 | comment | added | Wadim Zudilin | I am too optimistic, the magnitude is of size $\operatorname{const}/c^2$ (and a similar one for approximating $(d/b)^{1/2}$ by $ab/cd$) but at least it's clear that the examples could only come from continued fractions of quadratic irrationalities. | |
| Jul 23, 2010 at 4:37 | comment | added | Wadim Zudilin | The equality $a^2b^3-c^2d^3=1$ implies that the quadratic irrationality $(d/b)^{3/2}$ is "too well approximated" by the rational $a/c$. More precisely, $|(d/b)^{3/2}-a/c|<\operatorname{const}/c^4$. | |
| Jul 23, 2010 at 4:05 | comment | added | Steve Huntsman | To see this in MATLAB: for a=1:100,for b=1:100,z(a,b)=(a^2)*(b^3);end,end,z2=z(2:end,2:end);min(diff(unique(z2))) returns 4. | |
| Jul 23, 2010 at 4:04 | comment | added | Steve Huntsman | A computer search up to 100 for a and b shows a minimum difference of 4 amongst elements. | |
| Jul 23, 2010 at 3:57 | comment | added | Qiaochu Yuan | Have you tried a computer search? | |
| Jul 23, 2010 at 3:51 | history | asked | Ken Fan | CC BY-SA 2.5 |