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comment A miraculous number N
@GerryMyerson: Not quite. It's been proved that there is always a prime between consecutive cubes for large enough cubes, but we still haven't (!) ruled out the possibility that there are some integers $n>0$ with $\pi((n+1)^3)=\pi(n^3).$
Apr
20
comment Squarefree products of a class of primes
@WillJagy: Thanks -- post as an answer, maybe?
Apr
20
asked Squarefree products of a class of primes
Apr
19
revised Multiples of some set has density
fix mistake
Apr
19
comment Multiples of some set has density
@MatthewConroy: Yes, thanks!
Apr
17
revised Multiples of some set has density
it's already the best possible
Apr
17
asked Multiples of some set has density
Apr
16
revised Show that $\limsup_{x \to \infty} \frac{\pi(x)}{x/ \log x} \geqslant 1. $
tags
Apr
16
comment Are there arbitratily long runs of consecutive integers n that are NOT of the form $n = p^k$ or $n = 2p^k$ for some $k>0$ and $p$ an odd prime?
@GeoffreyCritzer: That's the Prime Number Theorem and it was the crowning achievement of 19th century analytic number theory.
Apr
16
comment Are there arbitratily long runs of consecutive integers n that are NOT of the form $n = p^k$ or $n = 2p^k$ for some $k>0$ and $p$ an odd prime?
@GeoffreyCritzer: All that matters for the asymptotic are primes (n/log n) and twice primes (0.5n/log n) which sum to 1.5n/log n and average 1.5/log n = 3/(2 log n). Squares, cubes, etc. are too rare to care about (as shown in my answer).
Apr
16
answered Are there arbitratily long runs of consecutive integers n that are NOT of the form $n = p^k$ or $n = 2p^k$ for some $k>0$ and $p$ an odd prime?
Apr
14
answered Sequence with Prime Numbers
Apr
13
comment Erdős Prime Sieve Conjecture
@JonMarkPerry: It's just a way to ensure that $p\pm1$ have no small prime factors except 2 and 3. Many other choices are possible.
Apr
13
answered Erdős Prime Sieve Conjecture
Apr
13
revised Is this of any real importance to the mathematical scientific community?
rm thx
Apr
13
comment Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?
It's stronger than Goldbach -- conceivably this cold fail even if Goldbach is true.Practically speaking both are extremely likely. $d=6$ is not very special -- it's a bit more likely to work since it's divisible by 2 and 3 (30 would be better, adding in also 5) but that's not a big deal. It might easily have had a small counterexample but it happens not to have -- 8,2,4 probably don't have any large counterexamples.
Apr
11
awarded  Nice Question
Apr
9
answered Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?
Apr
8
comment A generalization of Goldbach's conjecture?
@iadvd: I don't understand what you're asking. Feel free to post it as a new question (and ping me here, if you like).
Apr
7
comment A generalization of Goldbach's conjecture?
@iadvd: I think you are asking: Given some (large enough) even $n$, is there always a prime $p$ such that both $n-p$ and $n-p+6$ are prime?