Some vanishing sums involving binomial coefficients in the denominator

Identities involving binomial coeffcients usually arise in situations where counting is carried out in two different ways. For instance, some identities obtained by William Horrace [1] using probability theory turn out to be special cases of the Chu-Vandermonde identities. Here, we obtain some generalizations of the identities observed by Horrace and give different types of proofs; these, in turn, give rise to some other new identities. In particular, we evaluate sums of the form Pm j=0 (1) j j d (mj) (n+jj ) and deduce that they vanish when d is even and m = n > d=2. It is well-known [2] that sums involving binomial coeffcients can usually be expressed in terms of the hypergeometric functions but it is more interesting if such a function can be evaluated explicitly at a given argument. Identities such as the ones we prove could perhaps be of some interest due to the explicit evaluation possible. The papers [3], [4] are among many which deal with identities for sums where the binomial coeffcients occur in the denominator and we use similar methods here

Tiresome paths, water gates and Euler’s formula

A hallmark of mathematics is its power to look at seemingly different problems with the same eyes and find a common idea which resolves both. It is not surprising that the two problems we discuss here, about routes to be taken with various constraints and about watering fields, can both be treated using ideas from graph theor

Some Observations on Khovanskii\u27s Matrix Methods for extracting Roots of Polynomials

In this article we apply a formula for the n-th power of a 3×3 matrix (found previously by the authors) to investigate a procedure of Khovanskii’s for finding the cube root of a positive integer. We show, for each positive integer α, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to α 1/3 . We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a more general class of cubic equations, and, for each such cubic, obtain a sequence of rationals that converge to the real root of the cubic. We prove that Khovanskii’s method for finding the m-th (m ≥ 4) root of a positive integer works, provided a free parameter is chosen to satisfy a very simple condition. Finally, we briefly consider another procedure of Khovanskii’s, which also involves m×m matrices, for approximating the root of an arbitrary polynomial of degree m

Powers of a matrix and combinatorial identities

In this article we obtain a general polynomial identity in k variables, where k ≥ 2 is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a k × k matrix. Finally, we use these results to derive various combinatorial identities

Binary Cubic Forms and Rational Cube Sum Problem

The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; from the earlier works of Sylvester, Satg{\'e}, Selmer etc. and up to the recent work of Alp{\"o}ge-Bhargava-Shnidman. In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $1 \pmod {9d}$ as well as $-1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes

On a Conjecture of Chowlaet al.

AbstractWe prove some congruences for the numbers[formula]. In particular, we show that the numbers are congruent to 5 modulop3for any primep⩾5, thereby proving a conjecture of Chowlaet al.(J. Number Theory12(1980), 188–190)

Set theory revisited as easy as pie the principle of inclusion and exclusion – part 1

Recall the old story of two frogs from Osaka and Kyoto which meet during their travels. They want to share a pie. An opportunistic cat offers to help and divides the pie into two pieces. On finding one piece to be larger, she breaks off a bit from the larger one and gobbles it up. Now, she finds that the other piece is slightly larger; so, she proceeds to break off a bit from that piece and gobbles that up, only to find that the first piece is now bigger. And so on; you can guess the rest. The frogs are left flat