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Welcome to number theory! In this chapter we will see a bit of what number theory is about and why you might enjoy studying it. Carl Friedrich Gauss (1777–1855), one of the greatest mathematicians of all time, had this to say about number theory (which he called arithmetic):
number or groups of numbers in a discrete fashion, and in nested hierarchies re ecting various complexities, and even attach a symmetry group to each indi-vidual number. It is not that number theorists avoid the plane model, since it is also an important tool in much of number theory. It is just that the plane has a
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Anyone who has ever fallen in love will tell you it's the little things about the other person that matter. The silly in-jokes shared at the end of the day. The peculiarities of the other person's morning coffee ritual. The way he or she lets old paperbacks stack up on the bedside table. Such interrelated details come to define us. They trace the u...
The study of those sometimes subtle and far-reaching relationships is number theory, sometimes referred to as higher arithmetic. Number theorists scrutinize the properties of integers, the natural numbers you know as -1, -2, 0, 1, 2 and so forth. It's part theoretical and part experimental, as mathematicians seek to discover fascinating and even un...
What kind of relationships? Well, we actually categorize integers into different number types based on their relationships. There are, of course, odd numbers (1,3, 5 ), which cannot be divided evenly, and even numbers (2, 4, 6 ), which can. There are square numbers, produced by multiplying another number by itself. For instance, 2 x 2 = 4 and 3 x...
Modern number theory uses techniques from and contributes to areas across mathematics, including especially representation theory and algebraic geometry. Number theory also plays an important role in computer science, especially in public-key cryptography.
This is Number theory, a branch of mathematics that explores the properties and relationships of integers. The cool part is that anyone can join in, it is mostly basic operations like multiplication and addition! And along the way we will discover many interesting concepts, surprising relationships, and get to rub shoulders with great thinkers ...
Introduction to Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3 and so on, and there’s the negatives. Which one don’t you understand? After all, the mathematician G. H. Hardy wrote:
The Fundamental Theorem of Arithmetic: Every natural number n > 1 can be expressed in an essentially unique way as the product of prime numbers. By "essentially unique", I mean "counting different orderings of the primes as the same": 12 = 22 × 3 = 3 ×22, but I'm counting these products as essentially the same.
