## Posts Tagged 'Number Theory'

### Lecture sans paroles: the factors of M67

In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause.

### Prime Generating Formulae

The prime numbers have challenged and perplexed the greatest mathematicians for millennia. Shortly before he died, the brilliant Hungarian number theorist Paul Erdös said “it will be another million years, at least, before we understand the primes”.

A remarkable polynomial: Theorem 1 from Jones et al., 1976.

### Andrew Wiles wins 2016 Abel Prize

A recent post described the Abel Prize, effectively the Nobel Prize for Mathematics, and promised a further post when the 2016 winner was announced. This is the follow-up post [also at TM091, or search for “thatsmaths” at irishtimes.com].

Next Tuesday, HRH Crown Prince Haakon will present the Abel Medal to Sir Andrew Wiles at a ceremony in Oslo. The Abel Prize, comparable to a Nobel Prize, is awarded for outstanding work in mathematics. Wiles has won the award for his “stunning proof of Fermat’s Last Theorem” with his research “opening a new era in number theory”. Wiles’ proof made international headlines in 1994 when he cracked one of the most famous and long-standing unsolved problems in mathematics.

Pierre de Fermat, a French lawyer and amateur mathematician, stated the theorem in 1637, writing in the margin of a maths book that he had “a truly marvellous proof”. But for more than 350 years no proof was found despite the efforts of many of the most brilliant mathematicians.

### Ramanujan’s Astonishing Knowledge of 1729

Question: What is the connection between Ramanujan’s number 1729 and Fermat’s Last Theorem? For the answer, read on.

The story of how Srinivasa Ramanujan responded to G. H. Hardy’s comment on the number of a taxi is familiar to all mathematicians. With the recent appearance of the film The Man who Knew Infinity, this curious incident is now more widely known.

Result of a Google image search for “K3 Surface”.

Visiting Ramanujan in hospital, Hardy remarked that the number of the taxi he had taken was 1729, which he thought to be rather dull. Ramanujan replied “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

### Prime Number Record Smashed Again

Once again the record for the largest prime number has been shattered. As with all recent records, the new number is a Mersenne prime, a number of the form

Mp = 2p 1

where p itself is a prime. Participants in a distributed computing project called GIMPS (Great Internet Mersenne Prime Search) continue without rest to search for ever-larger primes of this form.

Most of the recent large primes have been found in the GIMPS project (for an earlier post on GIMPS, click Mersennery Quest. The project uses a search algorithm called the Lucas-Lehmer primality test, which is particularly suitable for finding Mersenne primes. The test, which was originally devised by Edouard Lucas in the nineenth century and extended by Derek Lehmer in 1930, is very efficient on binary computers.

We all know the festive carol The Twelve Days of Christmas. Each day, “my true love” receives an increasing number of gifts. On the first day there is one gift, a partridge in a pear tree. On the second, two turtle doves and another partridge, making three. There are six gifts on the third day, ten on the fourth, fifteen on the fifth, and so on.

Here is a Christmas puzzle: what is the total number of gifts over the twelve days? [TM083, or search for “thatsmaths” at irishtimes.com]

### The Tragic Demise of a Beautiful Mind

John Nash, who was the subject of the book and film A Beautiful Mind, won the Abel Prize recently. But his journey home from the award ceremony in Norway ended in tragedy [see this week’s That’s Maths column (TM069): search for “thatsmaths” at irishtimes.com].

Russell Crowe as John Nash in the movie A Beautiful Mind.

### Fermat’s Christmas Theorem

Albert Girard (1595-1632) was a French-born mathematician who studied at the University of Leiden. He was the first to use the abbreviations ‘sin’, ‘cos’ and ‘tan’ for the trigonometric functions.

Left: Albert Girard (1595-1632). Right: Pierre de Fermat (1601-1665)

### Waring’s Problem & Lagrange’s Four-Square Theorem

$\displaystyle \mathrm{num}\ = \square+\square+\square+\square$

Introduction

We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved numerous ingenious and beautiful results in partition theory.

More generally, additive number theory is concerned with the properties and behaviour of integers under addition. In particular, it considers the expression of numbers as sums of components of a particular form, such as powers. Waring’s Problem comes under this heading.

### Old Octonions may rule the World

This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton.

Multiplication table for octonions, of the form z=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]

Continue reading ‘Old Octonions may rule the World’

### Triangular Numbers: EYPHKA

The maths teacher was at his wits’ end. To get some respite, he set the class a task:

Add up the first one hundred numbers.

That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later dubbed the Prince of Mathematicians. Continue reading ‘Triangular Numbers: EYPHKA’

### Invention or Discovery?

Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:

Conjecture: Definitions are invented. Theorems are discovered.

The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’

### Breaking Weather Records

In arithmetic series, like 1 + 2 + 3 + 4 + 5 + … , each term differs from the previous one by a fixed amount. There is a formula for calculating the sum of the first N terms. For geometric series, like 3 + 6 + 12 + 24 + … , each term is a fixed multiple of the previous one. Again, there is a formula for the sum of the first N terms of such a series. Continue reading ‘Breaking Weather Records’

### The Unity of Mathematics

This week, That’s Maths in The Irish Times ( TM041  ) is about an ambitious program to unify mathematics.

Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have a comprehensive understanding of even a single field of mathematics: the web of knowledge grows so fast that no-one can master it all.

### The Langlands Program

An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects.

Canadian mathematician Robert Langlands, who formulated a series of far-reaching conjectures [image from Wikimedia Commons].

### The Prime Number Theorem

God may not play dice with the Universe, but something strange is going on with the prime numbers  [Paul Erdös, paraphrasing Albert Einstein]

The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural number greater than one can be expressed in a unique way as a product of primes. Continue reading ‘The Prime Number Theorem’

### Experiment and Proof

Many mathematicians spend their time proving results. The (very old) joke is that they are machines for turning coffee into theorems. A theorem is a statement that has been shown, by a sequence of irrefutable steps, to follow logically from a set of fundamental assumptions known as axioms.

These axioms themselves may be self-evident, or may simply be assumed to be true. Given this, the statement contained in a theorem is known with certainty to be true.

### The remarkable BBP Formula

Information that is declared to be forever inaccessible is sometimes revealed within a short period. Until recently, it seemed impossible that we would ever know the value of the quintillionth decimal digit of pi. But a remarkable formula has been found that allows the computation of binary digits starting from an arbitrary  position without the need to compute earlier digits. This is known as the BBP formula.
Continue reading ‘The remarkable BBP Formula’

### The Ups and Downs of Hailstone Numbers

Hailstones, in the process of formation, make repeated excursions up and down within a cumulonimbus cloud until finally they fall to the ground. We look at sequences of numbers that oscillate in a similarly erratic manner until they finally reach the value 1. They are called hailstone numbers.
Continue reading ‘The Ups and Downs of Hailstone Numbers’

### Amazing Normal Numbers

For any randomly chosen decimal number, we might expect that all the digits, 0, 1 , … , 9, occur with equal frequency. Likewise, digit pairs such as 21 or 59 or 83 should all be equally likely to crop up.  Similarly for triplets of digits. Indeed, the probability of finding any finite string of digits should depend only on its length. And, sooner or later, we should find any string. That’s “normal”!

### Prime Secrets Revealed

This week, That’s Maths in the Irish Times ( TM022 ) reports on two exciting recent breakthroughs in prime number theory.

The mathematics we study at school gives the impression that all the big questions have been answered: most of what we learn has been known for centuries, and new developments are nowhere in evidence. In fact, research in maths has never been more intensive and advances are made on a regular basis.

### Dis, Dat, Dix & Douze

How many fingers has Mickey Mouse? A glance at the figure shows that he has three fingers and a thumb on each hand, so eight in all. Thus, we may expect Mickey to reckon in octal numbers, with base eight. We use decimals, with ten symbols from 0 to 9 for the smallest numbers and larger numbers denoted by several digits, whose position is significant. Thus, 47 means four tens plus seven units.
Continue reading ‘Dis, Dat, Dix & Douze’

### Happy Pi Day 2013

Today, 14th March, is Pi Day. In the month/day format it is 3/14, corresponding to 3.14, the first three digits of π. So, have a Happy Pi Day. Larry Shaw of San Francisco’s Exploratorium came up with the Pi Day idea in 1988. About ten years later, the U.S. House of Representatives passed a resolution recognizing March 14 as National Pi Day.

Today is also the birthday anniversary of Albert Einstein, giving us another reason to celebrate. He was born on 14 March 1879, just 134 years ago today.
Continue reading ‘Happy Pi Day 2013’

### Chess Harmony

Long ago in the Gupta Empire, a great-but-greedy mathematician, Grababundel, presented to the Maharaja a new game that he had devised, called Chaturanga.

Thirty-two of the Maharaja’s subjects, sixteen dressed in white and sixteen in black, were assembled on a field divided into 64 squares. There were rajas and ranis, mahouts and magi, fortiers and foot-soldiers. Continue reading ‘Chess Harmony’

### Ramanujan’s Lost Notebook

In the Irish Times column this week ( TM010 ), we tell how a collection of papers of Srinivasa Ramanujan turned up in the Wren Library in Cambridge and set the mathematical world ablaze. Continue reading ‘Ramanujan’s Lost Notebook’

### The Root of Infinity: It’s Surreal!

Can we make any sense of quantities like “the square root of infinity”? Using the framework of surreal numbers, we can.

• In Part 1, we develop the background for constructing the surreals.
• In Part 2, the surreals are assembled and their amazing properties described.

### A Mersennery Quest

The theme of That’s Maths (TM008) this week is prime numbers. Almost all the largest primes found in recent years are of a particular form M(n) = 2n1. They are called Mersenne primes. The Great Internet Mersenne Prime Search (GIMPS) is aimed at finding ever more prime numbers of this form. Continue reading ‘A Mersennery Quest’

### The Beautiful Game

What is the most beautiful rectangular shape? What is the ratio of width to height that is most aesthetically pleasing? This question has been considered by art-lovers for centuries and one value appears consistently, called the golden ratio or Divine proportion. Continue reading ‘The Beautiful Game’