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It is unlikely. There are ways to compute the nth digit of certain numbers in certain bases (for example, pi in base 16) without having to compute the entire number, but in most situations, the numb …

This is implicit in Douglas Zare's comment to the problem as well as in domotorp's posted answer and the comments following, but I will make it explicit: one can substitute any slow growing sequence o …

EDIT: This answers the wrong question. > cn^3 is wanted, not > cp^3 . END EDIT To even hope for such a c, you will need to avoid the following kind of set: fix t with at least n factors, then for pr …

Short answer: I doubt you can do much better than trial factorization. However, for many composite numbers, the run time is often up to the size of the second largest prime factor, so you may actuall …

You are asking for consecutive runs of smooth numbers. I do not have the keyboard to spell Stormer with a stroke over the o, but http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem has information f …

One thing I would like to see more of is an analysis of the distribution of integers coprime to a large integer (totients of?) M. If M has k distinct prime factors, one can get M/2 as an upper bound …

Encouraged by Andre Henriques' comment (and not seeing a response which puts more constraints on the problem), I shall promote my comment to an answer. Consider an arbitrary convex polygon P in R^2 w …

Here is an insight. Look at the rectangular simplex formed in the positive orthant that is cut off by the plane H = wvec dot vvec, where wvec is your vector of weights. Your sum will be a weighted s …

You can rewrite the sum using prime gap notation. With $d_k=p_{k+1}-p_k$, the sum becomes $$ n^ap_n - \sum_{k=1}^{n-1} k^ad_k$$ and now you can use some knowledge of prime gaps to understand the las …

To add to Greg Kuperberg's observation, let n be given and p a prime so that log(factorial p) > n, where log is to the base 2. Then p! > (n choose k) for 0 <= k <= n, and so p! cannot divide (n choo …

Here is a partial answer. For those who want more information on this subject, Will Jagy has been kind enough to forward appropriate emails to me, to which I respond. I ask that you send requests a …

While waiting for Will Orrick to weigh in, I have an unprofessional opinion which says that this approach is unlikely to be more productive than many combinatorial approaches for finding D-optimal bin …

I have not looked at the literature, so the following may fail for some geometric reason of which I am unaware; this reflects my thinking, and I do not yet see how else to get the results. It should …