Testwiki:Reference desk/Archives/Mathematics/2015 September 17

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September 17

Dual of a trapezoid

A quadrilateral is the dual of a trapezoid if and only if... Georgia guy (talk) 16:28, 17 September 2015 (UTC)

You already asked that and I answered it; see Wikipedia:Reference desk/Archives/Mathematics/2015 May 5. If it's not clear maybe someone else has another POV. --RDBury (talk) 19:47, 17 September 2015 (UTC)

Numbers Fulfilling All Three Conditions

I am searching for numbers fulfilling all these three conditions: $\frac{(\genfrac{}{}{0ex}{}{2n}{n})}{(n+1)(n+2)}$ is not an integer, $n\ne 4k+2,$ and $n\ne 3k+1.$ I haven't been able to find any below 100,000. — 82.137.52.249 (talk) 21:07, 17 September 2015 (UTC)

You can drop the n+1 and just say $\frac{(\genfrac{}{}{0ex}{}{2n}{n})}{(n+2)}$ since C(2n,n) is always divisible by n+1 and n+1 and n+2 are relatively prime. Numbers such that n+2 does not divide C(2n,n) are listed in Template:Oeis so the question is equivalent to asking if there are any entries which are not congruent to 1, 2, 4, 6, 7, or 10 (mod 12). Equivalently, this is asking if n congruent to 0, 3, 5, 8, 9, or 11 (mod 12) implies n+2 | C(2n, n). Not a solution but hopefully helpful info. --RDBury (talk) 15:15, 18 September 2015 (UTC)