[seqfan] Re: Help needed... for proofs, Brocard's Conjecture, etc.

[Charles Greathouse]

Well, you can use A002386 here. Starting in the middle, if a number is less
than 1357201, then the gap between it and the prime preceding it is at most
118, and hence every gap at least 118*3+1 = 355 has the required four
primes. prime(n+1) is at least prime(n) + 2 = p + 2, so the corresponding
gap is at least (p+2)^2 - p^2 = 4p + 4. On the lower end we have 355 = 4p +
4 and so p = 87.75; on the upper end (p+2)^2 = 1357201 or p = 1162.9.... So
this shows that the desired inequality holds for primes in the range 88 < p
< 1162. Your next step can use the fact that the gap is at most 1462 for
numbers less than 4 * 10^18 to increase the bound to about 2 * 10^9. But
probably it would be just as easy to show this directly -- you only need
four primes, after all.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Tue, Dec 30, 2014 at 3:22 PM, Antti Karttunen <antti.karttunen at gmail.com>
wrote:

> Here are two things for which I need help right now (not just the end
> of the next year):
>
> In https://oeis.org/A050216 "Number of primes between (prime(n))^2 and
> (prime(n+1))^2, with a(0) = 2 by convention." it is told that
> Brocard's Conjecture states that for n >= 2, a(n) >= 4.
>
> See also http://en.wikipedia.org/wiki/Brocard%27s_conjecture
>
> Now the question: for up to which k it is proved (or "obvious") that
> A050216(n) >= k, for all n >= 2 ?
>
> For example, is it clear that https://oeis.org/A251723 "First
> differences of A054272, A250473 and A250474: a(n) = A054272(n+1) -
> A054272(n). " (and also "One less than A050216") is always positive (>
> 0) or even nonnegative (>= 0) ?
> I.e. that a sequence like https://oeis.org/A054272 is (genuinely) growing?
>
>
> ----------
>
> Secondly, can somebody prove at https://oeis.org/A251726
> "Numbers n > 1 for which there exists r <= gpf(n) such that r^k <=
> spf(n) and gpf(n) < r^(k+1) for some k >= 0, where spf and gpf
> (smallest and greatest prime factor of n) are given by A020639(n) and
> A006530(n). "
>
> my conjecture that:
> "If any n is in the sequence, then so is A003961(n)."
>
> where A003961 shifts the primes in the prime factorization of n one
> step towards larger primes,
> thus also spf(n) and gpf(n) will be replaced by the respective
> nextprimes. Note that as far as I see it, this is towards "unsafe
> direction", concerning the defining condition of A251726, because the
> "old r" doesn't necessarily work anymore, but a larger value is
> sometimes needed.
>
>
> In any case, I am myself absolutely lousy with any proofs involving
> limits, inequivalences or contradictions.
>
>
> Thanks in advance,
>
> Antti
>
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