hprime

Enthalpy thanks for all the wise advices ! Regarding Excel as a quick-and-dirty exploration tool, the differences computed are well over values where accuracy could be problematic. When I write "I have no computing power" I should have written more honestly "I have no programming skills", beyond my rusty BASIC from the 80's, and the XSLT I use daily in my job for data conversion, but this is way off anything useful here. Meanwhile, I've also set the question on the OEIS forum with some interesting results. http://list.seqfan.eu/pipermail/seqfan/2014-May/013065.html

@imatfaal I've no computing power at hand, just checked with MS Excel up to 2^1024 which is close to (3/2)^1750 or 10^308 where it breaks. If you define un as the increasing sequence of powers of 2 and 3/2 (1, 3/2, 2, 9/4, 27/8, 4 ... The difference between two consecutive terms of this sequence is growing with a law such that log(un+1 - un)/log(un) seems to converge towards 1. Which means (un+1 - un) is growing with the same order of magnitude than un itself. Of course my conjecture is based on a very small number of terms (less than 3000) but I take it very improbable that this law has sudden exceptions for larger values of n. And there again i'm pretty convinced it has something to do with Catalan's theorem or related conjectures, but I have not found a proof of equivalence so far.

@Enthalpy Thanks fo the hint, but I don't buy your proof, sorry. This was my first try at it The fact that the lower bound of |log(x) - log(y)| is 0 implies that the ratio x/y can be arbitrarily close to 1, that does not mean their difference can be arbitrary small. Trivial counter-example is log(n+1) - log(n) ... Actually, I'm pretty convinced that the lower bound in that case is NOT 0 in general and I'm searching a proof of that at least for 2 and 3/2. My hunch is that it has something to do with the Catalan's conjecture (which is not a conjecture any more). http://en.wikipedia.org/wiki/Catalan%27s_conjecture My conjecture is that having two powers of 2 and 3/2 arbitrarily close would break Catalan's theorem at some point. But arithmetic is tough

@imatfaal Your proof is correct and its quite trivial result I took for granted, sorry But the point is : if every value of |un(3) - uk(2)| is greater than 0 it does not imply the lowest bound of the set of such values is greater than 0. It is this lowest bound I'm interested in.

Thanks for the quick reaction, Unity+ But I think you misunderstood the question, or it was not clearly set. When I write : If p1 and p2 are distinct primes, is 0 the lower bound of |un(p1) - uk(p2)| ? I mean : the lower bound for a given couple (p1,p2), not over all values of (p1,p2) For the latter question, seems to me the answer is trivial and indeed 0, independently of the twin primes conjecture, we need only to know that there is an infinity of primes, which seems to be true Quick proof : For any e > 0, there exists some p such asu1(p) < 1 + e Then for all p' > p, u1(p') < 1 + e and then u1(p) - u1(p') < e Regarding your last question Could you clarify why you did "0.25 = (3/2)2 - 2"? This is the lowest |un(p1) - uk(p2)| value I could find "experimentally" for p1 = 2 and p2= 3/2, that is for n =1 and k= 2. My conjecture is that the two progressions don't come closer to each other for great values of n and k. But infinity is great, mainly towards the end

Hello all I'm new to this forum, I don't know if this topic or related ones have been discussed somewhere else, but a quick search did not bring me any relevant result. I'm investigating on geometric progressions with initial value 1 and ratio p/(p-1), where p is a prime number. For two different values of p, it's trivial that the two progressions will have no common term (apart of the initial 1). For example powers of 2, 3/2, 5/4 ... are all distinct. My question is : given two such progressions, can they contain terms arbitrarily close to each other? Put in more formally: Let un(p)=(p/(p-1))n, where p is a prime integer and n a positive integer (n >1) If p1 and p2 are distinct primes, is 0 the lower bound of |un(p1) - uk(p2)| ? I would be happy to have the answer even for the first prime numbers p1 = 2 and p2 = 3, that is : can one find a power of 3/2 arbitrarily close to a power of 2? My conjecture is that the lower bound is strictly greater than 0, and for the case of 2 and 3, the lower bound might be 0.25 = (3/2)2 - 2 Any related works that could point to a solution - or related open issues, there are many on prime numbers EDIT : "Harmonics" in the title might be not obvious. The original question comes from how we cope in (musical) harmony with the fact that 27 differs from (3/2)12, but not too much, so that according instruments is possible at all, dealing with the wolf interval in various ways or temperaments. (7/6)9 and 22 have less difference than the above pair, but harmonics based on the division of a string in seven parts are just ignored in occidental music. It led me to the general question of how close such harmonics can get ...