arXiv subjects
Publications and source records attributed to Robert J Betts.
Using a method we have utilized previously, namely through a finite power series expansion which also sometimes is known as the "radix polynomial" representation of an integer, we find an upper bound for a van der Waerden number that has a recurrence property.
Here we show that by expressing a van der Waerden number $W(r, k)$ by its radix polynomial representation, it not only is possible to locate each proper subset on $\mathbb{R}$ in which the van der Waerden number lies, but also to show that conditions exist for which the logarithm of the van der Waerden number necessarily is bounded above by the square of the number of terms $k$ in the arithmetic progression. Furthermore we also use the method to find a mathematical expression or formula for the ratio of two "consecutive" van der Waerden numbers of the kind $W(r, k)$, $W(r, k + 1)$.
We derive the asymptotic behaviors of $\log_{r}W(r, k)$ and $\log_{k}W(r, k)$, when $W(r, k)$ is a van der Waerden Number. We use the approach to consider the subsets on the real line in which $W(2, 7)$ might lie.
What is a least integer upper bound on van der Waerden number $W(r, k)$ among the powers of the integer $r$? We show how this can be found by expanding the integer $W(r, k)$ into powers of $r$. Doing this enables us to find both a least upper bound and a greatest lower bound on $W(r, k)$ that are some powers of $r$ and where the greatest lower bound is equal to or smaller than $W(r, k)$. A finite series expansion of each $W(r, k)$ into integer powers of $r$ then helps us to find also a greatest real lower bound on any $k$ for which a conjecture posed by R. Graham is true, following immediately as a particular case of the overall result.
We provide a way to modify and to extend a previously established inequality by P. Erdős, R. Graham and others and to answer a conjecture posed in the nineties by R. Graham, which bears on the lack of divisibility of the central binomial coefficient by three distinct, fixed odd primes. In fact the result will show by using an approach similar to their own which they proved for the case of two fixed odd primes, that the central binomial coefficient is not divisible infinitely often by three distinct and fixed odd primes. Therefore a generalization to more fixed odd primes than three but finite in number might be possible, at least if one is able to find some sufficient condition. The author hopes to answer this latter question in a subsequent paper.