Related papers: Open Conjectures on Congruences
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
This paper describes a method used to construct infinitely many probable counterexamples of the abc conjecture over the rational integers.
Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar Curvature and Convergence}. We have tried to survey all the progress towards these…
This is a collection of variants of Schanuel's conjecture and the known dependencies between them. It was originally written in 2007, and made available for a time on my webpage. I have been asked by a few people to make it available again…
A family of congruences interpolating between those of Wilson and Giuga is constructed. Several elementary results are established, in order to present a possible approach to establishing Giuga's conjecture.
Some thoughts are presented on the inter-relation between beauty and truth in science in general and theoretical physics in particular. Some conjectural procedures that can be used to create new ideas, concepts and results are illustrated…
Open-string theories may be related to suitable models of oriented closed strings. The resulting construction of ``open descendants'' is illustrated in a few simple cases that exhibit some of its key features.
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.
Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle…
We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form $(\root n\of{a_n})_{n\ge 1}$ or the form $(\root{n+1}\of{a_{n+1}}/\root n\of{a_n})_{n\ge1}$, where $(a_n)_{n\ge 1}$ is a…
In this paper, we survey some recent results on the Artin conjecture and discuss some aspects for the Artin conjecture.
Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We…
Some formulas and speculations are presented relative to integrable systems and quantum mechanics.
Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A…
Using algebraic transformations and equivalent reformulations we derive a number of new results from some earlier ones (by the author) in more accepted terms closely related to well-known conjectures of Bondy and Jung including a number of…
Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type…
Six conjectures on pairs of consecutive primes are listed in this paper, together with examples for each case.
This document contains a description of several of my papers, including remarks on history and connection with subsequent work. It also contains some new results and conjectures.
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…