## Additive Analogue of the Brocard – Ramanujan equation

I have mention Brocard’s problem in a previous post

https://kevinjdavis2013.wordpress.com/2014/12/30/brocards-problem-1897/

In this I noted the ‘proof’, or at least the scoping of the problem as only having a finite number of solutions, if the $abc$ conjecture is accepted.

In the three years since my post, computational studies have continued with ever larger numbers but still no counter examples have been found.

A rich seam of research has also studied variations of the Brocard – Ramanujan equation

$n!+1=m^2$

for example by replacing the factorial $n!$ with an analogue, $G_1G_2 \ldots G_k$; where the $G$ are Fibonnaci or Lucas numbers (the former has been completely solved).

I have been looking at an additive analogue of the Brocard – Ramanujan equation.

The factorial of $n$, $n! = n(n-1)(n-2) \ldots (2)(1)$, has an additive analogue $T_n=n+(n-1)+(n-2) + \ldots +(2)+(1)$, the triangular numbers $T_n$. The square number $m^2$ has an additive analogue $m+m$.

The Brocard – Ramanujan equation as an additive analogue is

$n+(n-1)+(n-2)+(2)+(1)+1=m+m$

Positive integer solutions of this variant can be found, and there are many(!); $(n,m)$ (Davis?) pairings of numbers from A174114 (sequence in the OEIS), the even central polygonal numbers divided by $2$, and A042963, the numbers congruent to $1mod 4$ or $2 mod 4$.

By noting that

$n+(n-1)+(n-2) + \ldots +(2)+(1) = \frac{n(n+1)}{2}$

it is simple to show that for the positive integer solutions of this Brocard – Ramanujan analogue, $n$ is mapped to $m$ by a quadratic curve.

Do other number theory conjectures yield similarly simple proofs on rewriting in an additive or multiplicative analogue form?

## Mathematics in Ancient Egypt

Below is a review I wrote for a CUP journal (The Mathematical Gazette) about a recently published text on mathematics in ancient Egypt

Mathematics in ancient Egypt, a contextual history by Annette Imhausen, pp 234, £21.05 (hard), ISBN 978-0-691-11713-3, Princeton University Press (2016)

This is an excellent book that falls mid-way between a specialised and general readership.

Abstractions made possible through mathematics, developed by the ancient Egyptian scribes, placed them in the seats of power.

The starting point for this book is the early prehistory and dynastic developments of number notation in both hieroglyphic and hieratic script (i.e. ‘printed and handwritten’ script). The basis of number systems in Egypt was 10, however, it was without place values and therefore with numerous hieroglyphic signs such as a coiled rope for 100 (the author suggests this may be derived from a measuring rope of 100 standard cubits used to establish field lengths for tax purposes). The earliest use of these numbers on ‘tags’ to record quantities of textiles with others on stela recording tributes. The book is richly illustrated throughout by clear line drawings and photographs of artefacts.

Developments in mathematics during the Old Kingdom, a time of political stability and (consequent) prosperity, occurred but sadly surviving written evidence is scarce. The author illustrates developments through secondary references such as the success of scribes in managing revenues and completing other administrative tasks. The author argues that the establishment of a lunar calendar at this time adds evidence to the scribes understanding of mathematics, not least due to the complexities of marrying observed solar and lunar cycles. The author also reports hints of the later developments in the analysis of gradients from findings at Saqqara.

Calendar from Kom Ombo (my photograph from Egypt)

Saqqara (my photograph from Egypt)

In the Middle Kingdom (2055-1650 BCE), following on from an intermediate period when climatic change led to famine, the breaking down of the central administration where scribes had developed and established their mathematics, the first mathematical texts are extant and appear to be derived from an educational context. They teach mathematics a scribe would need in daily work: calculating volumes of granaries, rations, ratios and solving, as the author calls them, ‘bread and beer problems’.

The extant mathematical texts from this period, as one would expect from such a scholarly work, are carefully selected to illustrate developments, and their mathematics clearly explained. Gems from this period include calculating the volume of a truncated pyramid, methods of arithmetic, fractions and in particular the 2/n tables that were crucial to the scribe’s calculations. It is in the use of these methods indicated in non-mathematical texts and tomb models that give an insight into the mathematics that existed; the author richly paints a picture of life in Egypt through examples of its use in calculating rations, and photographs of models from tombs with lines of scribes busying away writing alongside grain stores.

It is not surprising with the magnificent scale of architectural remains visible today, that much of the extant mathematical texts reveal efficient and detailed methods for scribes to use in metrology. The author explains sample methods from texts for calculations of area: perhaps of particular note are the methods for calculation of circular land areas based only on diameters (showing no knowledge of pi), and triangle area calculations that evidence no understanding of what we now call as the ‘rule of Pythagoras’. Contrast this with calculations in near contemporaneous China that clearly indicate both approximation to pi, and ‘Pythagoras’, were well known.

In the New Kingdom (1550-1069 BCE), the dynasties best-known today, the author notes it is surprising that more mathematics texts are not extant. During this period mathematical education remained at the heart of a trainee scribe’s studies. Along with the few scraps of mathematical texts the author points to tombs with the depictions of scribes with measuring ropes surveying in the fields.

One of the joys of translating early mathematical texts is the picture they paint of daily life. The author, through the use of mathematics for administrative purposes, reveals the richness of the lives of the elite with references to calculations involving precious stones, woods, ivories, fruits etc. Other essential, at the time, methods for constructing brick ramps and the transport of Obelisks were also known and carefully detailed in the book.

Luxor (my photograph from Egypt)

From this period there are many fragmentary texts that refer to mathematical education and one letter in particular selected by the author will have resonance for the readers of The Gazette: from (presumably) a parent to a student scribe it admonishes him to work hard saying ‘He who works in writing (mathematics) is not taxed, he has no dues (to pay)’.

Not surprisingly, due to the wealth of funerary objects that remain, the author details in them the evidence for use of mathematics, and for numerology in spells contained in the Book of the Dead.

The author deftly deals with speculations that abound by fanciful writers who are keen to project back in time the use of mathematical methods known today: theories that have been taken as ‘truth’ which do not stand up to rigorous studies of the evidence. However, the author does briefly speculate, rather convincingly, about Egyptian conceptualisation of space referencing well illustrated tomb carvings and geometrical problems.

The final chapter of the book details the Greco-Roman periods, a time the author concludes when more mathematical knowledge was transferred from Egypt to Greece than vice versa: that is one suggestion readers of The Gazette might use in conversations with colleagues in their Classics departments.

This was a time rich in extant mathematical texts that enable comparison with earlier methods of arithmetic. As in many other early mathematical texts they are presented as problems with worked solutions, several illuminating examples are included in this book. I was pleased to see the author refrained from trying to assign paths of transmission to specific mathematical knowledge as this is notoriously difficult and often vacuous: details the distinctive ‘pole against wall’ problems re surfaced at this time and their algorithmic solutions are carefully explained.

This book richly demonstrates how mathematics played a central role in ancient Egyptian culture. It clearly distinguishes the Egyptian mathematics from developments in Mesopotamia (where the sexagesimal number system was used) and places it as a distinct and valuable contribution to the development of mathematical knowledge. The result of ten years of research, this keenly priced book stands head and shoulders above others that purport to report on mathematics in ancient Egypt.

## Visual sheafs, presheafs, stalks and germs

Ok.

I know some mathematicians are a bit squeamish about visualizing concepts. However, attached is a sheet of interrelated diagrams that I find helpful to understand sheafs.

## Blowing up singularities

Commentary and examples, including one from Hartsthorne’s Algebraic Geometry. Trusting GCHQ are not bothered by the ‘blowing up’ in the title …

## Commentary on the Diagrams in The Red Book

In particular Spec (*) …

## Commentary on Mumford’s Red Book

My notes and commentary on the first couple of chapters of David Mumford’s: The Red Book of Varieties and Schemes.

## Visualizing affine algebraic varieties

An affine algebraic variety is an irreducible closed subset of ${A_k}^n$ (with the induced topology) as discussed in the previous two posts.

A non-constant polynomial $p(x,y)$ in the variables $x$ and $y$ defines a plane curve affine variety $Z(p) \subset {A_k}^2$. Below is the variety for the plane elliptic curve $p(x,y) := y^2-x^3+x$.

The set of four points $\{ (-2,-1),(-1,1),(1,-1),(1,2) \}$ in ${A_k}^2$ is an affine variety. It is the intersection of an ellipse $Z(x^2+y^2-xy-3)$ and a hyperbola $Z(3x^2-y^2-xy+2x+2y-3)$.

A quadratic is an affine algebraic variety defined by a single quadratic polynomial. In ${A_k}^2$ these are the plane conics; circles, ellipses, parabolas, and hyperbolas in $\mathbb{R}^2$. In ${A_k}^3$ these are; spheres, ellipsoids, paraboloids, and hyperboloids in $\mathbb{R}^3$.

Below are examples of the two paraboloid affine algebraic varieties, a hyperbolic paraboloid (a Pringle has this surface!) $Z(xy+z)$ and an elliptic paraboloid $Z(x^2+y^2-z)$.

Below are examples of two hyperboloid affine algebraic varieties, a hyperboloid of one sheet $Z(x^2-x+y^2+yz)$ and a hyperboliod of two sheets $Z(x^2+y^2-z^2+1)$.

Plane curves and quadratics defined by a single polynomial are called hyper-surfaces. Some affine algebraic varieties are intersections of hyper-surfaces.

Below is an example of an affine algebraic variety, the intersection of two quadratics $Z(x^2+y^2+z^2-1)$ and $Z(x^2-x+y^2)$ that is a `figure of eight’ curve in $\mathbb{R}^3$.

Irreducible varieties such as the plane curve $Z(y^2-x^3+x)$ in ${A_\mathbb{R}}^2$ and the surface $Z((x^2-y^2)^2-2x^2-2y^2-16z^2+1)$ in ${A_\mathbb{R}}^3$ are irreducible hyper-surfaces. These are visualised below; the first has components that are connected in Euclidean topology, the second has five components that meet at four singular points.

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