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I have studied the Lucas theorem and I encountered the following facts.

How to deduce the following facts from The Lucas theorem?

(1) If d, q > 1 are integers such that , $\binom{nd}{md}$ $\equiv$ $\binom{n}{m}$ (mod p) for every pair of integers n greater than or equal to m and m greater than or equal to 0, then d and q are powers of the same prime p.

(2) Let n and r be non-negative integers and p is greater than or equal to 5 be a prime, then $\binom{np}{rp}$ $\equiv$ $\binom{n}{r}$ (mod $p^3$). Where we set $\binom{n}{r}$ = 0, if n < r.

(3) Let N, R, n and r be non-negative integers and p is greater than or equal to 5 be a prime. Let n, r < p, then $\binom{Np^3+n}{Rp^3 + r}$ $\equiv$ $\binom{N}{R}$ $\binom{n}{r}$(mod $p^3$).

(4) If p is a prime and x is a positive integer and $p^x$ divides $\left\lfloor\frac{n}{p}\right\rfloor\, $ then, $p^x$ also divides $\binom{n}{p}$

I observed the above facts during my study on LUCAS Theorem. If any one can justify the all above four consequence problems, I am very glad and thankful to them. A waiting a good response on these interesting questions. Once gain thank you for this opportunity to post my questions.

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    You can find solutions [in this pdf](http://home.ecke$r$d.edu/~zhaoj/$r$esearch/ZhaoJNTBern.pdf)2012-09-13

0 Answers 0