The Sieve of Eratosthenes and a Partition of the Natural Numbers

The sieve of Eratosthenes is a method for finding all the prime numbers less than some maximum value $latex {M}&fg=000000$ by repeatedly removing multiples of the smallest remaining prime until no composite numbers less than or equal to $latex {M}&fg=000000$ remain. The sieve provides a means of partitioning the natural numbers. We examine this partition … Continue reading The Sieve of Eratosthenes and a Partition of the Natural Numbers →

The Golden Key to Riemann’s Hypothesis

The Riemann Hypothesis Perhaps the greatest unsolved problem in mathematics is to explain the distribution of the prime numbers. The overall ``thinning out'' of the primes less than some number $latex {N}&fg=000000$, as $latex {N}&fg=000000$ increases, is well understood, and is demonstrated by the Prime Number Theorem (PNT). In its simplest form, PNT states that … Continue reading The Golden Key to Riemann’s Hypothesis →

Elusive Transcendentals

The numbers are usually studied in layers of increasing subtlety and intricacy. We start with the natural, or counting, numbers $latex {\mathbb{N} = \{ 1, 2, 3, \dots \}}&fg=000000$. Then come the whole numbers or integers, $latex {\mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \}}&fg=000000$. All the ratios of these (avoiding division … Continue reading Elusive Transcendentals →

A Remarkable Sequence in OEIS

The On-Line Encyclopedia of Integer Sequences (OEIS), launched in 1996, now contains 360,000 entries. It attracts a million visits a day, and has been cited about 10,000 times. It is now possible for anyone in the world to propose a new sequence for inclusion in OEIS. The goal of the database is to include all … Continue reading A Remarkable Sequence in OEIS →

A Topological Proof of Euclid’s Theorem

Theorem (Euclid): There are infinitely many prime numbers. Euclid's proof of this result is a classic. It is often described as a proof by contradiction but, in fact, Euclid shows how, given a list of primes up to any point, we can find, by a finite process, another prime number; so, the proof is constructive. … Continue reading A Topological Proof of Euclid’s Theorem →

Numbers Without Ones: Chorisenic Sets

There is no end to the variety of sets of natural numbers. Sets having all sorts of properties have been studied and many more remain to be discovered. In this note we study the set of natural numbers for which the decimal digit 1 does not occur. Google Translate on my mobile phone gives the … Continue reading Numbers Without Ones: Chorisenic Sets →

Amusical Permutations and Unsettleable Problems

In a memorial tribute in the Notices of the American Mathematical Society (Ryba, et al, 2022), Dierk Schleicher wrote of how he convinced John Conway to publish a paper, ``On unsettleable arithmetical problems'', which included a discussion of the Amusical Permutations. This paper, which discusses arithmetical statements that are almost certainly true but likely unprovable, … Continue reading Amusical Permutations and Unsettleable Problems →

A Puzzle: Two-step Selection of a Digit

Here is a simple problem in probability. (1) Pick a number k between 1 and 9. Assume all digits are equally likely. (2) Pick a number m in the range from 1 to k. What is the probability distribution for the number m? A graph of the probability distribution is shown in the figure here. … Continue reading A Puzzle: Two-step Selection of a Digit →

Ford Circles & Farey Series

American mathematician Lester Randolph Ford Sr. (1886--1967) was President of the Mathematical Association of America from 1947 to 1948 and editor of the American Mathematical Monthly during World War II. He is remembered today for the system of circles named in his honour. For any rational number $latex {p/q}&fg=000000$ in reduced form ($latex {p}&fg=000000$ and … Continue reading Ford Circles & Farey Series →

Summing the Fibonacci Sequence

The Fibonacci sequence must be familiar to anyone reading this. We define it by means of a second-order recurrence relation, $latex \displaystyle F_{n+1} = F_{n-1} + F_n \,. \ \ \ \ \ (1)&fg=000000$ and two initial values, $latex {F_0 = 0}&fg=000000$ and $latex {F_1 = 1}&fg=000000$. This immediately yields the well-known sequence $latex \displaystyle … Continue reading Summing the Fibonacci Sequence →

Closeness in the 2-Adic Metric

When is 144 closer to 8 than to 143? The usual definition of the norm of a real number $latex {x}&fg=000000$ is its modulus or absolute value $latex {|x|}&fg=000000$. We measure the ``distance'' between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric $latex {\rho(x,y) = |x-y|}&fg=000000$ … Continue reading Closeness in the 2-Adic Metric →

Fields Medals presented at IMC 2022

Every four years, at the International Congress of Mathematicians, the Fields Medal is awarded to two, three, or four young mathematicians. To be eligible, the awardees must be under forty years of age. For the chosen few, who came from England, France, Korea and Ukraine, the award, often described as the Nobel Prize of Mathematics, … Continue reading Fields Medals presented at IMC 2022 →

Goldbach’s Conjecture and Goldbach’s Variation

Goldbach's Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude. The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to … Continue reading Goldbach’s Conjecture and Goldbach’s Variation →

Fairy Lights on the Farey Tree

The rational numbers $latex {\mathbb{Q}}&fg=000000$ are dense in the real numbers $latex {\mathbb{R}}&fg=000000$. The cardinality of rational numbers in the interval $latex {(0,1)}&fg=000000$ is $latex {\boldsymbol{\aleph}_0}&fg=000000$. We cannot list them in ascending order, because there is no least rational number greater than $latex {0}&fg=000000$. However, there are several ways of enumerating the rational numbers. The … Continue reading Fairy Lights on the Farey Tree →

Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes

In last week's post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found --- even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal … Continue reading Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes →

Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes

We define an extension of parity from the integers to the rational numbers. Three parity classes are found --- even, odd and none. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The natural numbers $latex {\mathbb{N}}&fg=000000$ split nicely into two subsets, the odd and even numbers $latex \displaystyle … Continue reading Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes →

Gaussian Primes

We are all familiar with splitting natural numbers into prime components. This decomposition is unique, except for the order of the factors. We can apply the idea of prime components to many more general sets of numbers. The Gaussian integers are all the complex numbers with integer real and imaginary parts, that is, all numbers … Continue reading Gaussian Primes →

Number Partitions: Euler’s Astonishing Insight

In 1740, French mathematician Philippe Naudé wrote to Leonhard Euler asking in how many ways a positive integer can be written as a sum of distinct numbers. In his investigations of this, Euler established the theory of partitions, for which he used the term partitio numerorum. Many of Euler's results in number theory involved divergent … Continue reading Number Partitions: Euler’s Astonishing Insight →

Set Density: are even numbers more numerous than odd ones?

In pure set-theoretic terms, the set of even positive numbers is the same size, or cardinality, as the set of all natural numbers: both are infinite countable sets that can be put in one-to-one correspondence through the mapping $latex {n \rightarrow 2n}&fg=000000$. This was known to Galileo. However, with the usual ordering, $latex \displaystyle \mathbb{N} … Continue reading Set Density: are even numbers more numerous than odd ones? →

The Spine of Pascal’s Triangle

We are all familiar with Pascal's Triangle, also known as the Arithmetic Triangle (AT). Each entry in the AT is the sum of the two closest entries in the row above it. The $latex {k}&fg=000000$-th entry in row $latex {n}&fg=000000$ is the binomial coefficient $latex {\binom{n}{k}}&fg=000000$ (read $latex {n}&fg=000000$-choose-$latex {k}&fg=000000$), the number of ways of … Continue reading The Spine of Pascal’s Triangle →

All Numbers Great and Small

Is space continuous or discrete? Is it smooth, without gaps or discontinuities, or granular with a limit on how small a distance can be? What about time? Can time be repeatedly divided into smaller periods without any limit, or is there a shortest interval of time? We don’t know the answers. There is much we … Continue reading All Numbers Great and Small →

The Simple Arithmetic Triangle is full of Surprises

Pascal’s triangle is one of the most famous of all mathematical diagrams, simple to construct and yet rich in mathematical patterns. These can be found by a web search, but their discovery by study of the diagram is vastly more satisfying, and there is always a chance of finding something never seen before [TM212 or search … Continue reading The Simple Arithmetic Triangle is full of Surprises →

Goldbach’s Conjecture: if it’s Unprovable, it must be True

The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of … Continue reading Goldbach’s Conjecture: if it’s Unprovable, it must be True →

Derangements and Continued Fractions for e

We show in this post that an elegant continued fraction for $latex {e}&fg=000000$ can be found using derangement numbers. Recall from last week's post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position. The number of … Continue reading Derangements and Continued Fractions for e →

Arrangements and Derangements

Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer --- about 37% whatever the number of students --- emerges from the theory of … Continue reading Arrangements and Derangements →

Will RH be Proved by a Physicist?

The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, $latex {\Re(s) = 1/2}&fg=000000$. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya … Continue reading Will RH be Proved by a Physicist? →

The p-Adic Numbers (Part 2)

Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930. Hensel is best known for his introduction of the … Continue reading The p-Adic Numbers (Part 2) →

The p-Adic Numbers (Part I)

The motto of the Pythagoreans was ``All is Number''. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers $latex {\mathbb{N}}&fg=000000$, and ratios of these, the positive rational numbers $latex {\mathbb{Q}^{+}}&fg=000000$. It came as a great shock that the diagonal of a … Continue reading The p-Adic Numbers (Part I) →

Think of a Number: What are the Odds that it is Even?

Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime? Since the set $latex {\mathbb{N}}&fg=000000$ of natural numbers is infinite, there are difficulties in assigning probabilities to subsets of $latex {\mathbb{N}}&fg=000000$. We require the probability … Continue reading Think of a Number: What are the Odds that it is Even? →

The Ever-growing Goals of Googology

In 1920, a kindergarten class was asked to describe the biggest number that they could imagine. One child proposed to “write down digits until you get tired”. A more concrete idea was to write a one followed by 100 zeros. This number, which scientists would express as ten to the power 100, was given the … Continue reading The Ever-growing Goals of Googology →

The Online Encyclopedia of Integer Sequences

Suppose that, in the course of an investigation, you stumble upon a string of whole numbers. You are convinced that there must be a pattern, but you cannot find it. All you have to do is to type the string into a database called OEIS --- or simply ``Slone's'' --- and, if the string is … Continue reading The Online Encyclopedia of Integer Sequences →

Bang! Bang! Bang! Explosively Large Numbers

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is $latex \displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 &fg=000000$ which is approximately $latex {8\times 10^{53}}&fg=000000$. The number of atoms in the universe is estimated to be about $latex {10^{80}}&fg=000000$. When we consider permutations of large sets, even … Continue reading Bang! Bang! Bang! Explosively Large Numbers →

The “extraordinary talent and superior genius” of Sophie Germain

When a guitar string is plucked, we don't see waves travelling along the string. This is because the ends are fixed. Instead, we see a standing-wave pattern. Standing waves are also found on drum-heads and on the sound-boxes of violins. The shape of a violin strongly affects the quality and purity of the sound, as … Continue reading The “extraordinary talent and superior genius” of Sophie Germain →

Spiralling Primes

The prime numbers have presented mathematicians with some of their most challenging problems. They continue to play a central role in number theory, and many key questions remain unsolved. Order and Chaos The primes have many intriguing properties. In his article ``The first 50 million prime numbers'', Don Zagier noted two contradictory characteristics of the … Continue reading Spiralling Primes →

Closing the Gap between Prime Numbers

Occasionally, a major mathematical discovery comes from an individual working in isolation, and this gives rise to great surprise. Such an advance was announced by Yitang Zhang six years ago. [TM161 or search for “thatsmaths” at irishtimes.com]. After completing his doctorate at Purdue in 1991, Zhang had great difficulty finding an academic position and worked at various … Continue reading Closing the Gap between Prime Numbers →

Massive Collaboration in Maths: the Polymath Project

Sometimes proofs of long-outstanding problems emerge without prior warning. In the 1990s, Andrew Wiles proved Fermat's Last Theorem. More recently, Yitang Zhang announced a key result on bounded gaps in the prime numbers. Both Wiles and Zhang had worked for years in isolation, keeping abreast of developments but carrying out intensive research programs unaided by … Continue reading Massive Collaboration in Maths: the Polymath Project →

Multiple Discoveries of the Thue-Morse Sequence

It is common practice in science to name important advances after the first discoverer or inventor. However, this process often goes awry. A humorous principle called Stigler's Law holds that no scientific result is named after its original discoverer. This law was formulated by Professor Stephen Stigler of the University of Chicago in his publication … Continue reading Multiple Discoveries of the Thue-Morse Sequence →

Really, 0.999999… is equal to 1. Surreally, this is not so!

The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory. In school we learn that … Continue reading Really, 0.999999… is equal to 1. Surreally, this is not so! →

Random Numbers Plucked from the Atmosphere

Randomness is a slippery concept, defying precise definition. A simple example of a random series is provided by repeatedly tossing a coin. Assigning “1” for heads and “0” for tails, we generate a random sequence of binary digits or bits. Ten tosses might produce a sequence such as 1001110100. Continuing thus, we can generate a … Continue reading Random Numbers Plucked from the Atmosphere →

Listing the Rational Numbers III: The Calkin-Wilf Tree

The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the Farey Sequences. A second, related, approach gives us the Stern-Brocot Tree. Here, we introduce … Continue reading Listing the Rational Numbers III: The Calkin-Wilf Tree →

Listing the Rational Numbers II: The Stern-Brocot Tree

The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. But it is not obvious how to construct a list that is sure to contain every rational number precisely once. In a previous post we described the Farey Sequences. Here we examine another, related, approach. The Stern-Brocot Tree We … Continue reading Listing the Rational Numbers II: The Stern-Brocot Tree →

Listing the Rational Numbers: I. Farey Sequences

We know, thanks to Georg Cantor, that the rational numbers --- ratios of integers --- are countable: they can be put into one-to-one correspondence with the natural numbers. How can we make a list that includes all rationals? For the present, let us consider rationals in the interval $latex {[0,1]}&fg=000000$. It would be nice if … Continue reading Listing the Rational Numbers: I. Farey Sequences →

Hardy’s Apology

Godfrey Harold Hardy's memoir, A Mathematician's Apology, was published when he was 63 years old. It is a slight volume at just 90 pages, but is replete with interesting observations and not a few controversial opinions. After 78 years, it is still in print and is available in virtually every mathematics library. Though many of … Continue reading Hardy’s Apology →

Kaprekar’s Number 6174

The Indian mathematician D. R. Kaprekar spent many happy hours during his youth solving mathematical puzzles. He graduated from Fergusson College in Pune in 1929 and became a mathematical teacher at a school in Devlali, north-east of Mumbai. Kaprekar is remembered today for a range of curious mathematical patterns that he discovered. The best known … Continue reading Kaprekar’s Number 6174 →

More on Moduli

We wrote last week on modular arithmetic, the arithmetic of remainders. Here we will examine a few other aspects of this huge subject. Modular arithmetic was advanced by Gauss in his Disquisitiones Arithmeticae. In this system, number wrap around when they reach a point known as the modulus. Numbers that differ by a multiple of … Continue reading More on Moduli →

Fractions of Fractions of Fractions

Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number $latex {x}&fg=000000$ can be expanded as a continued fraction: $latex \displaystyle x = a_0 + \cfrac{1}{ a_1 … Continue reading Fractions of Fractions of Fractions →

It’s as Easy as Pi

Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see TM120 or search for “thatsmaths” at irishtimes.com]. The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You … Continue reading It’s as Easy as Pi →

A Remarkable Pair of Sequences

The terms of the two integer sequences below are equal for all $latex {n}&fg=000000$ such that $latex {1<n<777{,}451{,}915{,}729{,}368}&fg=000000$, but equality is violated for this enormous value and, intermittently, for larger values of $latex {n}&fg=000000$. Hypercube Tic-Tac-Toe The simple game of tic-tac-toe, or noughts and crosses, has been generalized in several ways. The number of cells in … Continue reading A Remarkable Pair of Sequences →