Erdos-Straus with 4 terms - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:32:18Z http://mathoverflow.net/feeds/question/105786 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105786/erdos-straus-with-4-terms Erdos-Straus with 4 terms Aeryk 2012-08-29T02:43:07Z 2012-08-29T07:52:20Z <p>The <a href="http://en.wikipedia.org/wiki/Erdos-Straus_conjecture" rel="nofollow">Erdos-Straus conjecture</a> states that any fraction of the form $\frac{4}{n}$ can be decomposed as an Egyptian fraction with just 3 terms. In related research, I've recently come across conditions on $n$ (e.g. if $n$ has factorization of such and such a form) so that $\frac{k}{n}$ can be decomposed as an Egyptian fraction with $k$ terms. This isn't strong enough to prove Erdos-Straus, but I was wondering if there were already results like this out there or if this is new. It seems like most of what I read examines the 3-term decompositions. Are 4-term decompositions trivial to come by?</p> http://mathoverflow.net/questions/105786/erdos-straus-with-4-terms/105801#105801 Answer by Christian Elsholtz for Erdos-Straus with 4 terms Christian Elsholtz 2012-08-29T07:46:43Z 2012-08-29T07:52:20Z <p>Sums of 4 (or generally $k$) unit fractions are by no means trivial.</p> <p>There is a general criterion to Y. Rav (On the representation of rational numbers as a fixed sum of unit fractions. J. Reine Angew. Math. 222 (1966): 207-213.)</p> <p><a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00218186X&amp;IDDOC=253620" rel="nofollow">http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00218186X&amp;IDDOC=253620</a></p> <p>The equation $\frac{m}{n}= \frac{1}{x_1} + \cdots + \frac{1}{x_k}$<br> is certainly soluble for $m \leq k$, but for $m>k$ the same type of problems arise that one has for the Erdos-Straus equation.</p> <p>One can expect that for fixed $m$ and fixed $3 \leq k &lt; m$, there is some finite bound $N_{m,k}$ such that $n>N_{m,k}$ admits a solution. But this is an open problem.</p> <p>One can prove that for "almost all" $n \leq N$. The strongest version of "almost all", and a discussion of the parametrization of such equations is in C. Elsholtz, Sums of $k$ unit fractions, Trans. Amer. Math. Soc. 353 (2001), 3209-3227 </p> <p><a href="http://www.ams.org/journals/tran/2001-353-08/S0002-9947-01-02782-9/" rel="nofollow">http://www.ams.org/journals/tran/2001-353-08/S0002-9947-01-02782-9/</a></p>