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Robert J. Betts

Publications and source records attributed to Robert J. Betts.

5 recordsLinked to original sources
arXiv2015

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 $W(r, k) < r^{k^{2}}$

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 Graham, B. Rothschild and Joel Spencer back in the nineteen nineties, that the van der Waerden number with $r$ colorings and with arithmetic progressions of $k$ terms, has a certain upper bound. Our proof does not need the application of double induction, constructive methods of proof or combinatorics, as applied to sets of integers that contain some van der Waerden number as an element. The proof instead derives from certain \emph{a priori} knowledge that is known about any positive integer when the integer is large. The mathematical methods we use are easily accessible by those whose field of specialization lies outside of combinatorial number theory, such as discrete mathematics, computational complexity, elementary number theory or analytic number theory.

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arXiv2012

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

Does the $n^{th}$ 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 than or besides monotonicity. We show such a property holds for which the sequence of $n^{th}$ roots does converge, even if one has no a priori knowledge as to whether the sequence is monotone or not. We show also the $n^{th}$ root of the diagonal Ramsey number can be expressed as a product of two factors, the first being a known convergent sequence and the second being an absolutely convergent infinite series. One also can express it where one product is convergent and the other has all its values from a uniformly convergent complex function holomorphic within the unit disc on the complex plane. Our motivation solely is to prove the conjecture as a problem in search of a solution, not to establish some deep theory about graphs. A second question is: If the limit exists what is it? At the time of this writing the understanding is the proofs sought need not be constructive. Here we show by nonconstructive proofs that the $n^{th}$ root of the diagonal Ramsey number converges to a finite limit. We also show that the limit of the $j^{th}$ root of the diagonal Ramsey number is two, where positive integer $j$ depends upon the Ramsey number.

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arXiv2012

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 $\varepsilon$

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 combinatorics as applied to sets of integers that contain some van der Waerden number as an element. Rather we obtain the result solely by exploiting certain properties of any integer greater than one that is divisible by another integer. Our mathematical methods are easily accessible by those whose field of specialization lies outside of combinatorial number theory, such as discrete mathematics, elementary number theory or analytic number theory.

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arXiv2011

Deriving from an Assumption a semistable elliptic Curve for \(x_{0}^{k} + y_{0}^{l} = z_{0}^{m}\)

With a simple transformation of the three exponents the generalized Fermat equation can be put into the same form as the Fermat equation. When it is rewritten into this new altered form any real solutions to the altered equation equal a subset of real solutions to the Fermat equation. This result, along with a paradigm previously established by G. Frey and Y. Hellegouarch, can be used to show that the generalized Fermat equation has no solutions in coprime integers. In fact we prove one can derive an elliptic curve over $\mathbb{Q}$ for primes (p \geq 3), from the assumption that (x_{0}^{k} + y_{0}^{l} = z_{0}^{m}), (k, l, m \geq 3), has coprime solutions in integers. The curve then can be used to show the unsolvability of this equation. Our aim is not to arrive at very deep results. Rather the goal in this paper is to apply prior results obtained by G. Frey, K. Ribet and A. Wiles, to show the lack of solutions to the generalized Fermat equation.

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arXiv2010

A uniformly convergent Series for $ζ(s)$ and closed Formulas, that include Catalan Numbers

There exists an infinite series of ratios by which one can derive the Riemann zeta function $ζ(s)$ from Catalan numbers and central binomial coefficients which appear in the terms of the series. While admittedly the derivation is not deep it does indicate some combinatorial aspect to the Riemann zeta function. But we actually do find also four additional new closed formulas, which include a formula by which one can compute $ζ(s)$ for a countably infinite number of discrete positive values for $s$ where the formula contains Catalan numbers not in infinite series. The Riemann zeta function has applications in physics, such as in computations related to the Casimir effect. Our result indicates a link between the Riemann zeta function, combinatorics, Catalan numbers, the central binomial coefficient and the content of a hypersphere, under certain conditions.

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