Let 
be prime and
 |  |  |
(1)
|
 |  |  |
(2)
|
then
 |
(3)
|
This is proved in Fine (1947).
This theorem is the underlying reason that the binomial coefficient 
mod 2 can be computed using bitwise operations AND(NOT(
),
),
giving the Sierpiński sieve.
Lucas Correspondence Theorem
See also
Lucas Correspondence, Sierpiński SieveExplore with Wolfram|Alpha
References
Fine, N. J. "Binomial Coefficients Modulo a Prime." Amer. Math. Monthly 54, 589-592, 1947.Referenced on Wolfram|Alpha
Lucas Correspondence TheoremCite this as:
Weisstein, Eric W. "Lucas Correspondence Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LucasCorrespondenceTheorem.html