Bailey, D. F. Some binomial coefficient congruences. (English) Zbl 0732.11010 Appl. Math. Lett. 4, No. 4, 1-5 (1991). 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. Reviewer: B.Garrison (San Diego) Cited in 2 Documents MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities 11A07 Congruences; primitive roots; residue systems 05A10 Factorials, binomial coefficients, combinatorial functions Citations:JFM 10.0139.04 PDFBibTeX XMLCite \textit{D. F. Bailey}, Appl. Math. Lett. 4, No. 4, 1--5 (1991; Zbl 0732.11010) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.