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Some binomial coefficient congruences. (English) Zbl 0732.11010

É. Lucas [Sur les congruences des nombres Eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. Fr. 6, 49-54 (1878; JFM 10.0139.04)] has proved that if p is prime, \(n,r,n_ 0,r_ 0\) are nonnegative integers, \(n_ 0<p\), \(r_ 0<p\), then \(\left( \begin{matrix} np+n_ 0\\ rp+r_ 0\end{matrix} \right)\equiv \left( \begin{matrix} n\\ r\end{matrix} \right)\left( \begin{matrix} n_ 0\\ r_ 0\end{matrix} \right)(mod p)\). It is now proved that if p is prime, n,r,i are nonnegative integers, \(0<i<p\), then \[ \left( \begin{matrix} np\\ rp+i\end{matrix} \right)\equiv (r+1)\left( \begin{matrix} n\\ r+1\end{matrix} \right)\left( \begin{matrix} p\\ i\end{matrix} \right)(mod p^ 2). \] It is also shown that if \(p\geq 5\) is prime, \(i>0\), \(0\leq n\leq m\), \(0\leq k<p\), then \[ \left( \begin{matrix} mp^ 2\\ np^ 2+kp+i\end{matrix} \right)\equiv (n+1)\left( \begin{matrix} m\\ n+1\end{matrix} \right)\left( \begin{matrix} p^ 2\\ kp+i\end{matrix} \right)(mod p^ 3). \] The first proof proceeds by induction on n and the second by induction on m.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
05A10 Factorials, binomial coefficients, combinatorial functions

Citations:

JFM 10.0139.04
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References:

[1] Fine, N. J., Binomial coefficients modulo a prime, American Math Monthly, 54, 589-592 (1947) · Zbl 0030.11102
[2] Lucas, Édouard, Sur les congruences des nombres Eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. France, 6, 49-54 (1878) · JFM 10.0139.04
[3] Bailey, D. F., Two \(p^3\) variations of Lucas’ Theorem, Journal of Number Theory, 35, 208-215 (1990) · Zbl 0704.11001
[4] Dickson, L. E., (History of the Theory of Numbers, Vol. I (1952), Chelsea Publishing Co: Chelsea Publishing Co N.Y), 73
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