Robert J. Betts

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20102016

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8 papers · 1 filter

math.NT2016

Two Upper Bounds for both the van der Waerden Numbers $W (r, k + 1)$ and $W (r + 1, k)$ , that show the Existence of a Recurrence Relation

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…

math.NT2016

The Asymptotic Behaviors of $lo g_{r} W (r, k)$ and $lo g_{k} W (r, k)$ , when $W (r, k)$ is a van der Waerden Number

Robert J Betts

We derive the asymptotic behaviors of $lo g_{r} W (r, k)$ and $lo g_{k} W (r, k)$ , when $W (r, k)$ is a van der Waerden Number. We use the approach to consider the subsets on the real l…

math.NT2012

A nonconstructive Proof to show the Convergence of the $n^{t h}$ root of diagonal Ramsey Number $r (n, n)$

Robert J. Betts

Does the $n^{t h}$ root of the diagonal Ramsey number converge to a finite limit? The answer is yes. A sequence can be shown to converge if it satifies convergence conditions other…

math.NT2012

The Expansion of each van der Waerden number $W (r, k)$ into Powers of $r$ , when $r$ is the Number of Integer Colorings, determines a greatest lower Bound for all $k$ such that $W (r, k) < r^{k^{2}}$

Robert J. Betts

Every positive integer greater than a positive integer $r$ can be written as an integer that is the sum of powers of $r$ . Here we use this to prove the conjecture posed by Ronald G…

math.NT2012

A simple Proof that $r^{k^{2}}$ is an upper Bound on each van der Waerden Number $W (r, k)$ , for each $k$ bounded below by a certain positive real Number $ε$

Robert J. Betts

Here we answer a conjecture by Ron Graham about getting finer upper bounds for van der Waerden numbers in the affirmative, but without the application of double induction or combin…

math.NT2010

Lack of Divisibility of $(N 2 N)$ by three fixed odd primes infinitely often, through the Extension of a Result by P. Erdős, et al

Robert J Betts

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…

Two Upper Bounds for both the van der Waerden Numbers W(r,k+1) and W(r+1,k), that show the Existence of a Recurrence Relation

The Asymptotic Behaviors of logr​W(r,k) and logk​W(r,k), when W(r,k) is a van der Waerden Number

A nonconstructive Proof to show the Convergence of the nth root of diagonal Ramsey Number r(n,n)

The Expansion of each van der Waerden number W(r,k) into Powers of r, when r is the Number of Integer Colorings, determines a greatest lower Bound for all k such that W(r,k)<rk2

A simple Proof that rk2 is an upper Bound on each van der Waerden Number W(r,k), for each k bounded below by a certain positive real Number ε

Lack of Divisibility of (N2N​) by three fixed odd primes infinitely often, through the Extension of a Result by P. Erdős, et al

Two Upper Bounds for both the van der Waerden Numbers $W (r, k + 1)$ and $W (r + 1, k)$ , that show the Existence of a Recurrence Relation

The Asymptotic Behaviors of $lo g_{r} W (r, k)$ and $lo g_{k} W (r, k)$ , when $W (r, k)$ is a van der Waerden Number

A nonconstructive Proof to show the Convergence of the $n^{t h}$ root of diagonal Ramsey Number $r (n, n)$

The Expansion of each van der Waerden number $W (r, k)$ into Powers of $r$ , when $r$ is the Number of Integer Colorings, determines a greatest lower Bound for all $k$ such that $W (r, k) < r^{k^{2}}$

A simple Proof that $r^{k^{2}}$ is an upper Bound on each van der Waerden Number $W (r, k)$ , for each $k$ bounded below by a certain positive real Number $ε$

Lack of Divisibility of $(N 2 N)$ by three fixed odd primes infinitely often, through the Extension of a Result by P. Erdős, et al