Charles
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 1d 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 $\pi((n+1)^3)=\pi(n^3).$ Apr20 comment Squarefree products of a class of primes @WillJagy: Thanks -- post as an answer, maybe? Apr20 asked Squarefree products of a class of primes Apr19 revised Multiples of some set has density fix mistake Apr19 comment Multiples of some set has density @MatthewConroy: Yes, thanks! Apr17 revised Multiples of some set has density it's already the best possible Apr17 asked Multiples of some set has density Apr16 revised Show that $\limsup_{x \to \infty} \frac{\pi(x)}{x/ \log x} \geqslant 1.$ tags Apr16 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. Apr16 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). Apr16 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? Apr14 answered Sequence with Prime Numbers Apr13 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. Apr13 answered Erdős Prime Sieve Conjecture Apr13 revised Is this of any real importance to the mathematical scientific community? rm thx Apr13 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. Apr11 awarded Nice Question Apr9 answered Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)? Apr8 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). Apr7 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?