English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For all positive integers n, we prove the following divisibility properties:$$(2n+3){2n\choose n} | 3{6n\choose 3n}{3n\choose n}, and (10n+3){3n\choose n}| 21{15n\choose 5n} {5n\choose n}.$$ This confirms two recent conjectures ofZ.-W. Sun. Some similar divisibility properties are given. Moreover, we showthat, for all positive integers m and n, the product $am{am+bm-1\chooseam}{an+bn\choose an}$ is divisible by m+n. In fact, the latter result can begeneralized to the q-binomial coefficients and q-integers case, whichgeneralizes the positivity of q-Catalan numbers. We also propose severalrelated conjectures.

Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun