Saturday, 28 July 2007

A graph on irrationals where p is adjacent to q if p^q or q^p is rational.

The answers are that |E(G)|=2aleph0 and that every vertex of G has degree aleph0.



Proof:
The positive real solutions to xy=2 form a curve of cardinality 2aleph0, and at most aleph0 of these have x or y rational, so |E(G)|=2aleph0. (It cannot be larger, because this is also the number of pairs of vertices.)



For ginG with g>1, the set of positive real numbers x such that gx is a prime number contains at most one rational number since if there were two, then we would obtain an equation pq=p with p,p distinct primes and q rational, which is impossible by unique factorization. Thus g has infinite degree. On the other hand, the degree of g is at most countable since gx=q and xg=q have at most one solution x each for each qinmathbfQ. Finally, the same arguments apply when g<1, with reciprocals of primes in place of primes.



Remark: The same statements hold for G except that one should exclude the vertex 1.

No comments:

Post a Comment