So we can pair off numbers, so that a number in the pair is the inverse of the other number in the pair.

This is evident from archaeological artifactssuch as a 10,year-old bone from the Congo region of Africa with tally marks scratched upon it—signs of an unknown ancestor counting something.

Platonem ferunt didicisse Pythagorea omnia "They say Plato learned all things Pythagorean". We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in Number theory.

While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. Classical Greece and the early Hellenistic period[ edit ] Further information: You will require a little knowledge in advance, but not much: This function has a special property.

It is certain that an understanding of numbers existed in ancient Mesopotamia, EgyptChina, and India, for tablets, papyri, and temple carvings from these early cultures have survived. We need a little property of primes to help us later. The Number theory remainder theorem appears as an exercise [16] in Sunzi Suanjing 3rd, 4th or 5th century CE.

Given this viewpoint, it is not surprising that the Pythagoreans attributed Number theory properties to certain numbers. They knew only a single amicable pair: For those prone to number mysticism, such a phenomenon must have seemed like magic.

Page 1 of 4. Despite such isolated results, a general theory of numbers was nonexistent. I hope this Number theory given you a flavour of what Number Theory is about; there are numerous books available that continue to develop the theory, and large numbers of olympiad problems you might like to tackle with your new knowledge!

By arithmetic he meant, in part, theorising on number, rather than what arithmetic or number theory have come to mean.

In this article we shall look at some elementary results in Number Theory, partly because they are interesting in themselves, partly because they are useful in other contexts for example in olympiad problemsand partly because they will give you a flavour of what Number Theory is about.

Ancient Greek mathematics Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period. There is also some numerical mysticism in Chinese mathematics, [note 5] but, unlike that of the Pythagoreans, it seems to have led nowhere.

The Fundamental Theorem of Arithmetic: The Greek philosopher Nicomachus of Gerasa flourished c. By "essentially unique", I mean "counting different orderings of the primes as the same": We can use this result and induction to prove the following very important theorem: When does this happen?

You can find an explanation of all of this in the article called Modular Arithmetic. A Babylonian tablet known as Plimpton c. For instance, they attached significance to perfect numbers —i. At first sight, it might seem totally unclear how one could go about proving this, but there is a beautiful and simple proof that I shall outline now.

This certainly reveals a degree of number theoretic sophistication in ancient Babylon. But this is absurd. Eusebius, PE X, chapter 4 mentions of Pythagoras: Pythagoras According to tradition, Pythagoras c. His philosophy enshrined number as the unifying concept necessary for understanding everything from planetary motion to musical harmony.

But a number could be the inverse of itself. The Pythagorean tradition spoke also of so-called polygonal or figurate numbers.Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory.

These categories reflect the methods used to address problems concerning the integers. Number Theory Warmups If numbers aren't beautiful, we don't know what is. Dive into this fun collection to play with numbers like never before, and start unlocking the connections that are the foundation of Number Theory.

Rated 5 out of 5 by Fumi from An Introduction to Number Theory I am a retired petroleum engineer (age 78). By the way I am a female. I had a very satisfying career, vs.

fast changing and difficult time for the young people. Elementary number theory is taught in discrete mathematics courses for computer scientists; on the other hand, number theory also has applications to the continuous in numerical analysis.

As well as the well-known applications to cryptography, there are also applications to many other areas of mathematics. [Chap. 1] What Is Number Theory?

8 Sums of Squares II. Which numbers are sums of two squares? It often turns out that questions of this sort are easier to answer ﬁrst for primes, so we ask. In this article we shall look at some elementary results in Number Theory, partly because they are interesting in themselves, partly because they are useful in other contexts (for example in olympiad problems), and partly because they will give you a flavour of what Number Theory is about.

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