the topology of arithmetic progressions of primes - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:48:15Z http://mathoverflow.net/feeds/question/104059 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104059/the-topology-of-arithmetic-progressions-of-primes the topology of arithmetic progressions of primes Vidit Nanda 2012-08-05T22:42:20Z 2012-09-01T19:57:25Z <p>The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner that will be made precise below. </p> <h2>Setup</h2> <p>Consider a nested family of simplicial complexes $K(p)$ indexed by prime $p \in \mathbb N$ defined as follows: </p> <ol> <li>the vertices are all primes less than or equal to $p$, and</li> <li>insert a $d$-simplex ($d \geq 2$) spanning $d+1$ vertices if and only if they constitute an arithmetic progression. Of course, one must also insert all faces, and faces-of-faces etc. so that the defining property of a simplicial complex is preserved.</li> </ol> <p>For instance, $K(7)$ has the vertices $2,3,5,7$ and a single $2$-simplex $(3,5,7)$ along with all its faces. $K(11)$ has all this, plus the vertex $11$ and the simplex $(3,7,11)$. The edge $(3,7)$ already exists so only the other two need to be added. Thus, the fact that $(3,7)$ occurs in two arithmetic progressions bounded by $11$ is encoded by placing the corresponding edge in the boundary of two simplices.</p> <h2>Question</h2> <p>Has someone already defined and studied this complex? What I am mostly interested in is</p> <blockquote> <p>How does the homology of $K(p)$ change with $p$?</p> </blockquote> <p>It is easy to check that the only interesting homology is in dimensions $\leq 1$. If it helps, here are -- according to home-brew computations -- the statistics for the first few primes (Betti 0 and 1 over $\mathbb{Z}_2$). I've already confirmed that the sequence of Betti-1's is not in Sloane's online encyclopedia of integer sequences. If an intermediate K[p] is missing in the list, that means that the homology is the same as that for the previous prime.</p> <h2>Update 1##</h2> <p>Zack pointed out an error in the previous computations, so here are the betti numbers with that error fixed. I have also removed "2" from the vertex set since the only contribution of that vertex to the homology is incrementing all the $0$ dimensional betti numbers by +1.</p> <p>K [3]: 1 0<br> K [5]: 2 0<br> K [7]: 1 0<br> K [13]: 2 0<br> K [17]: 2 1<br> K [19]: 1 2<br> K [23]: 1 4<br> K [31]: 1 6<br> K [37]: 2 6<br> K [43]: 1 7<br> K [53]: 1 8<br> K [59]: 1 9<br> K [61]: 1 10<br> K [67]: 1 12<br> K [71]: 1 17<br> K [73]: 1 20<br> K [79]: 1 23<br> K [83]: 1 26<br> K [89]: 1 31<br> K [97]: 1 32<br> K [101]: 1 35<br> K [103]: 1 41<br> K [107]: 1 43<br> K [109]: 1 47<br> K [113]: 1 53<br> K [127]: 1 58<br> K [131]: 1 62<br> K [137]: 1 67<br> K [139]: 1 73<br> K [149]: 1 78 </p> <h2>Update 2</h2> <p>I've just finished running the homology computations (over $\mathbb{Z_2}$) for all primes less than $30,000$, and <a href="https://dl.dropbox.com/u/74531549/prime_30k_betti.txt" rel="nofollow">this text file</a> contains the resulting Betti numbers. Once such data is available for control experiments, such as Cramer numbers or primes which are $1$ mod 4, I will put up those text files as well. Also, I am no longer confident that higher homology will not appear, so here is an auxiliary question:</p> <blockquote> <p>Is it true that the higher homology groups of $K(p)$ are trivial?</p> </blockquote>