A binomial sum related to Wolstenholme's theorem
نویسندگان
چکیده
منابع مشابه
A Binomial Sum Related to Wolstenholme’s Theorem
A certain alternating sum u(n) of n + 1 products of two binomial coefficients has a property similar to Wolstenholme’s theorem, namely u(p) ≡ −1 (mod p3) for all primes p ≥ 5. However, this congruence also holds for certain composite integers p which appear to always have exactly two prime divisors, one of which is always 2 or 5. This phenomenon will be partly explained and the composites in qu...
متن کاملA Digital Binomial Theorem
In this article, we demonstrate how the Binomial Theorem in turn arises from a one-parameter generalization of the Sierpinski triangle. The connection between them is given by the sum-of-digits function, s(k), defined as the sum of the digits in the binary representation of k (see [1]). For example, s(3) = s(1·2+1·2) = 2. Towards this end, we begin with a well-known matrix formulation of Sierpi...
متن کاملA Treatise on the Binomial Theorem
OF THE DISSERTATION A treatise on the binomial theorem by PATRICK DEVLIN Dissertation Director: Jeff Kahn This dissertation discusses four problems taken from various areas of combinatorics— stability results, extremal set systems, information theory, and hypergraph matchings. Though diverse in content, the unifying theme throughout is that each proof relies on the machinery of probabilistic co...
متن کاملA Generalization of the Digital Binomial Theorem
In [9], the author introduced a digital version of this theorem where the exponents appearing in (1) are viewed as sums of digits. To illustrate this, consider the binomial theorem for N = 2: (x + y) = xy + xy + xy + xy. (2) It is easy to verify that (2) is equivalent to (x+ y) = xy + xy + xy + xy, (3) where s(k) denotes the sum of digits of k expressed in binary. For example, s(3) = s(1 · 2 + ...
متن کاملBinomial Coefficient–harmonic Sum Identities Associated to Supercongruences
We establish two binomial coefficient–generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo p (k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene’s ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2009
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2009.05.010