arXiv subjects

Philip Gibbs

Publications and source records attributed to Philip Gibbs.

12 recordsLinked to original sources

arXiv2026

Multiple Solutions to Exponential Diophantine Equations of Ramanujan-Nagell type: $Cz^2 = D + A.B^n$

Ramanujan found five solutions to the exponential Diophantine equation $z^2+7= 2^n$ where $z$ and $n$ are positive integers, and posed the problem of determining whether there are any more. Nagell was the first to prove that there were not. It is natural to then ask if there are other similar Diophantine equations with multiple solutions. In particular, equations of the form $Cz^2=D+A.B^n$ are known as equations of Ramanujan-Nagell type. Three examples are known with six solutions. Summaries of multiple solutions for different cases are presented. In particular the equation $z^2=277665.17^6+34^n$ is found to have four solutions and is conjectured to be the only such non-trivial equation of Ramanujan-Nagell type with four solutions when $A=C=1$ and $B$ is not a power of 2.

math.GM↗

arXiv2024

Novel required properties of, and efficient algorithms to seek, perfect cuboids

We present a novel approach to the age-old question of whether perfect cuboids exist. Our approach consists of two new computer search algorithms, arising from the analysis of "perfect plinths" reported by one of us recently, that are much more efficient than any prior algorithm of which we are aware. Using this approach, we also (i) identify some new two-parameter families of edge cuboids, and (ii) show that large proportions of aspect ratios of faces or internal rectangles cannot be present in a perfect cuboid, a property that was previously reported only for faces with the aspect ratio 4:3.

math.GM↗

arXiv2018

An Upper Bound for Lebesgue's Covering Problem

A covering problem posed by Henri Lebesgue in 1914 seeks to find the convex shape of smallest area that contains a subset congruent to any point set of unit diameter in the Euclidean plane. Methods used previously to construct such a covering can be refined and extended to provide an improved upper bound for the optimal area. An upper bound of 0.8440935944 is found.

math.MG↗

arXiv2015

The Lebesgue Universal Covering Problem

In 1914 Lebesgue defined a "universal covering" to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pal, Sprague and Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen's last reduction was microsopic: he claimed to remove an area of $6 \cdot 10^{-18}$, but we show that he actually removed an area of just $8 \cdot 10^{-21}$. In the following, with the help of Greg Egan, we find a new, smaller universal covering with area less than $0.8441153$. This reduces the area of the previous best universal covering by a whopping $2.2 \cdot 10^{-5}$.

math.MG↗

arXiv2014

A New Slant on Lebesgue's Universal Covering Problem

Lebesgue's universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypotheses based on the conjectures. A new upper bound of 0.844113 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous best results by removing corners from a regular hexagon at a small slant angle to the edges of the dodecahedron used before. Simulations indicate that the minimum area for a convex universal cover is likely to be around 0.84408.

math.MG↗

arXiv2013

An Acataleptic Universe

John Wheeler advocated the principle that information is the foundation of physics and asked us to reformulate physics in terms of bits. The goal is to consider what we know already and work out a new mathematical theory in which space, time and matter are secondary. An application of the converse of Noether's second theorem to the holographic principle implies that physics must have an underlying hidden symmetry with degrees of symmetry that match physical degrees of freedom in order to account for the huge redundancy of information in the interior of a black-hole. I have been working on a theory that builds infinite dimensional symmetries using layers of quantisation from information as suggested by Wheeler's contemporary Carl von Weizsäcker. Necklace Lie algebras are the mathematical objects and iterated integration can be used to show how a continuum background can emerge from their structure. The logic suggests the conclusion that wheeler was right when he proclaimed "It from Bit"

physics.hist-ph↗

arXiv2011

A Universe Programmed with Strings of Qubits

It has been suggested that reality works like a quantum computer, but such claims are just words if they are not backed up by sound mathematics. In pursuit of the fundamental equations I look to string theory where physicists led by Mike Duff have noticed useful connections between the quantum gravity of black holes and quantum information theory. By building on my earlier work on universal symmetry in string theory and using links between elliptic curves and hyperdeterminants, I find intriguing clues that these connections may be deep as well as useful. Ultimately any theory of the foundations of physics must explain why there are four forces and three generations of fermions. In string theory this would be a consequence of the choice of vacua. If a consistent formulation of string theory constructed from quantum bits can be found, it may be possible to understand the vast landscape of possibilities better and reverse engineer the program that codes our universe.

hep-th↗

arXiv2010

Elliptic Curves and Hyperdeterminants in Quantum Gravity

Hyperdeterminants are generalizations of determinants from matrices to multi-dimensional hypermatrices. They were discovered in the 19th century by Arthur Cayley but were largely ignored over a period of 100 years before once again being recognised as important in algebraic geometry, physics and number theory. It is shown that a cubic elliptic curve whose Mordell-Weil group contains a Z2 x Z2 x Z subgroup can be transformed into the degree four hyperdeterminant on a 2x2x2 hypermatrix comprising its variables and coefficients. Furthermore, a multilinear problem defined on a 2x2x2x2 hypermatrix of coefficients can be reduced to a quartic elliptic curve whose J-invariant is expressed in terms of the hypermatrix and related invariants including the degree 24 hyperdeterminant. These connections between elliptic curves and hyperdeterminants may have applications in other areas including physics.

math.GM↗

arXiv2001

Diophantine Quadruples and Cayley's Hyperdeterminant

A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Such Diophantine sets have been used to construct high rank elliptic curves. Here I demonstrate a link with Cayley's hyperdeterminants which provides a fruitful generalisation of Diophantine quadruples.

math.NT↗

arXiv1999

A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples

Diophantine quadruples are sets of four distinct positive integers such that the product of any two is one less than a square. All known examples belong to an infinite set which can be constructed recursively. Some observations on these regular solutions are presented. In particular we see how factors of the four numbers satisfy relations which generalise the Stern-Brocot Tree. There is also an update on the search for rational Diophantine sextuples with five new examples.

math.NT↗

arXiv1999

Some Rational Diophantine Sextuples

A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm used to find them is described. A classification of quadruples and quintuples with examples and statistics is also given.

math.NT↗

arXiv1998

A White Hole Model of the Big Bang

A model of the universe as a very large white hole provides a useful alternative inhomogeneous theory to pit against the homogeneous standard FLRW big bang models. The white hole would have to be sufficiently large that we can fit comfortably inside the event horizon at the present time, so that the inhomogeneities of space-time are not in contradiction with current observational limits. A specific Lemaitre-Tolman model of a spherically symmetric non-rotating white hole with a few adjustable parameters is investigated. Comparison of calculated anisotropy in the Hubble flow and the CMB against observational limits constrain the parameter space. A Copernican principle would require that we are not too near the centre of the white hole. As an additional constraint this predicts a value of Omega between 0.9999 and 1.

gr-qc↗