Subject: binomial coefficients for 2-adic integers Date: Fri, 1 Jun 2001 09:44:23 -0400 From: "Ken Monks" -------------------------------------------- Hello all, Let n,k be positive integers. We know that Lucas' theorem says that the binomial coefficient (nCk) is odd iff no 0 appears above a 1 when you write the binary representation of n above the binary representation of k. Now let n and k be 2-adic integers. We can still write the 2-adic representation of n above the 2-adic representation of k and compare digits in the same way to get a function of two 2-adic variables that returns 1 if no 0 appears above a 1 and returns 0 otherwise. Is there is a natural definition of a "2-adic binomial coefficient", B, so that for any 2-adics integers n,k, nBk is a 2-adic integer which is odd iff the generalized Lucas condition holds? By "natural" I mean that nBk=nCk when k is a nonnegative integer and that many of the usual binomial coefficient identities still hold. References? Thanks! KEN ----------------------------------------- Ken Monks - Professor of Mathematics University of Scranton Scranton, PA 18510 email: mailto:monks@scranton.edu web: http://www.scranton.edu/~monks -----------------------------------------