A prime number is a natural number greater than 1 with no divisors other than 1 and itself. According to Euclid's theorem, there are infinitely many prime numbers, so there is no largest prime.

On October 12, 2024, the GIMPS (Great internet Mersenne Prime Search) discovered 52^nd Mersenne prime, the number 2^136279841 - 1, which has 41,024,320 digits and is now the largest known prime number.

Here we present that number printed, in all its digits, in a single volume.

Note: due to the number of digits in the number and the limitations of the physical book, the print is in a very small font.

Discover how mathematical sequences abound in our natural world in this definitive exploration of the geography of the cosmos

You need not be a philosopher or a botanist, and certainly not a mathematician, to enjoy the bounty of the world around us. But is there some sort of order, a pattern, to the things that we see in the sky, on the ground, at the beach? In A Beginner's Guide to Constructing the Universe, Michael Schneider, an education writer and computer consultant, combines science, philosophy, art, and common sense to reaffirm what the ancients observed: that a consistent language of geometric design underpins every level of the universe, from atoms to galaxies, cucumbers to cathedrals. Schneider also discusses numerical and geometric symbolism through the ages, and concepts such as periodic renewal and resonance. This book is an education in the world and everything we can't see within it.

Contains numerous b&w photos and illustrations.

2017 Reprint of 1941 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition software. G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. This 'Apology', written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'.

One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, particularly, his own special field of number theory.

Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.
In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.
Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory.
Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples. Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated.

Uncover the Hidden Wisdom of Sacred Geometry: Your Gateway to Divine Alignment and Transformation

In a world where challenges often appear insurmountable, where climate change serves as an urgent reminder of a deeper crisis, there exists a beacon of hope-a profound wisdom concealed within the timeless realm of Sacred Geometry.

Sacred Geometry: The Universal Language of Divine Alignment beckons both newcomers and seasoned seekers on a vivid voyage into the heart of reality's concealed enigmas. This book is your key to unlocking the secrets vital for our survival and spiritual evolution.

Embark on a journey where you will:

1. Unveil Universal Wisdom: Shift your energy and awareness by immersing yourself in the profound truths concealed within the very fabric of time and space, governed by the geometric dance of the Sun, Moon, and Earth.

2. Explore Your Inner Architecture: Acquire fresh insights into the intricate structure of your existence and discover how divine alignment holds the power to transform life's challenges into opportunities for growth.

3. Ascend to Higher Dimensions: Sacred geometry serves as the ancient key that unlocks gateways to elevated realms of understanding and consciousness.

4. Tools for Inner Peace: Learn the art of constructing and utilizing sacred geometry tools to attain emotional equilibrium, conquer stress, and craft a life of abundance and well-being.

5. Blueprint for Survival: Delve deep within the recesses of your heart center to open a portal to the wisdom of your soul. Transcend the illusion of separation and unlock your true potential to not merely survive but thrive.

With over four decades of dedicated expertise, Gail and Gregory Hoag have pioneered groundbreaking technologies that have transformed the lives of tens of thousands across the globe with their teaching and tools. They have harnessed the power of divine archetypes to amplify higher dimensional energies and elevate the human experience, catalyzing personal and spiritual growth towards the path of higher consciousness, well-being, and profound connection.

But number theory has also recently become of great practical importance - in the area of cryptography, where the security of your credit card, and indeed of the nation's defence, depends on a result concerning prime numbers that dates back to the 18th century. Recent years have witnessed other spectacular developments, such as Andrew Wiles's proof of 'Fermat's last theorem' (unproved for over 250 years) and some exciting work on prime numbers. In this Very Short Introduction Robin Wilson introduces the main areas of classical number theory, both ancient and modern. Drawing on the work of many of the greatest mathematicians of the past, such as Euclid, Fermat, Euler, and Gauss, he situates some of the most interesting and creative problems in the area in their historical context.

ABOUT THE SERIES:
The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. W. R. Dedekind. The first presents Dedekind's theory of the irrational number-the Dedekind cut idea-perhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational numbers, which had been used on an intuitive basis since Greek times. This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis.
The second essay is an attempt to give a logical basis for transfinite numbers and properties of the natural numbers. It examines the notion of natural numbers, the distinction between finite and transfinite (infinite) whole numbers, and the logical validity of the type of proof called mathematical or complete induction.
The contents of these essays belong to the foundations of mathematics and will be welcomed by those who are prepared to look into the somewhat subtle meanings of the elements of our number system. As a major work of an important mathematician, the book deserves a place in the personal library of every practicing mathematician and every teacher and historian of mathematics. Authorized translations by Vooster V. Beman.

A look at one of the most exciting unsolved problems in mathematics today

Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.

The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep--and often very mystifying--mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.

From zero to infinity, this entertaining hardback guide opens up a new world of knowledge based on the magic of numbers.

Numbers have occupied our thoughts since man first realized he had not one opposable thumb but two. And from simple enumeration they have grown to be the most important and universal language there is.

The Book of Numbers highlights the dominant role that numbers play in everyday life, as well as exploring how numbers and number systems evolved, and delving into the mysteries of mankind's most powerful numbers:

- What are the top-ten One Hit Wonders?
- What's so magnificent about 7?
- Why is 13 unlucky?
- And who exactly is beast number 666?

From algebra to astrology, music to mythology, from religion to recreation and from science to superstition, The Book of Numbers embraces this infinitely broad subject and puts it all in order-beginning with 0.