Problems and results on consecutive integers and prime factors of binomial coefficients (Q1068125)

The author lists a number of interesting problems on consecutive integers and prime factors of binomial coefficients. He outlines the progress on these problems. Quite a few of the progress on these problems is due to the author. Also the author has encouraged the progress of work on these and many other problems. For example the ({\$} 500) problem \[ (1/\phi (n))\sum^{\phi (n)-1}_{i=1}(u_{i+1}-u_i)^{\alpha}\ll (n/\phi (n))^{\alpha} \] for \(\alpha =2\) proposed by the author has been solved recently in fact for \(\alpha\geq 2\) by H. L. Montgomery and R. C. Vaughan. Here \(n\geq 3\), \(1=u_1<u_2<...<u_{\phi (n)}=n-1\) are the integers less than and coprime to n.