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WindowTitle->1. What Is Number Theory? | Introduction to Number Theory
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Cell["\<\
Number theory studies numbers and their properties. The properties can range \
from very natural and easy ones to the most sophisticated.\
\>", "Text",
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Mathematicians use different methods and tools to study numbers. Hence, \
number theory has different subdivisions, which we\[CloseCurlyQuote]ll \
address in this lesson.\
\>", "Text",
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Cell["\<\
The story of number theory begins over two millennia ago, and yet, it is a \
living subject. Newer and newer aspects and sides are discovered in number \
theory as mathematics and the world progresses. Number theory gets a new \
boost with each new era in science.\
\>", "Text",
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Cell["\<\
The latest discoveries in number theory and its importance are connected with \
electronic computers. Because modern computers can handle incredibly large \
numbers, we\[CloseCurlyQuote]re now doing what our ancestors couldn\
\[CloseCurlyQuote]t even dream about!\
\>", "Text",
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 " is one of the oldest and most intuitive fields of mathematics. It studies \
numbers ",
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Number theory aims to discover fascinating and unexpected patterns in numbers.\
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How many ways are there to make $2.00 from only nickels and quarters?\
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millennium, but there were great strides in number theory being made by \
Chinese and Indian mathematicians. ",
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When Renaissance-era Europe was first exposed to the mathematics developed in \
the Islamic world, they did not consider number theory a serious subject. \
Between 1400 and 1650, there were significant advancements in the fields of \
geometry, algebra and probability, as well as logarithms and analytic \
geometry being discovered. Despite these advancements, people still perceived \
number theory as a recreational hobby.\
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theorists, and Fermat\[CloseCurlyQuote]s last theorem, one of mathematics\
\[CloseCurlyQuote] most famous statements."
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Fermat rarely supplied proofs, so many did not see the value in number \
theory. The invention of calculus near the end of the seventeenth century was \
an additional hurdle for mainstream acceptance of the field of number theory. \
Given the practicality of calculus, scholars in many disciplines rushed to \
solve real-world problems with calculus. They viewed number theory as too \
pure and divorced from practical problems in physics, astronomy and \
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most prolific and influential mathematicians ever, and he eventually \
popularized number theory and Fermat\[CloseCurlyQuote]s works. When Euler \
began to work on number theory, he was initially indifferent to it, as most \
mathematicians were, but his correspondence with ",
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 " changed his mind. Goldbach was so enthusiastic about number theory and \
Fermat\[CloseCurlyQuote]s work that it started to rub off on Euler. After \
Euler\[CloseCurlyQuote]s conversion to number theory, the wider mathematical \
community finally started taking the field seriously."
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Cell["\<\
In contrast to Fermat rarely publishing proofs of his work, Euler spent 50 \
years publishing over a thousand pages of research and proofs. Many of his \
proofs proved Fermat\[CloseCurlyQuote]s theorems. He published his proof of \
Fermat\[CloseCurlyQuote]s little theorem in 1736. There were only three pairs \
of amicable numbers known when Euler started working on them, but he found 58 \
new pairs through his work.\
\>", "Text",
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Cell["\<\
Once Euler had legitimized the field of number theory, progress was fast.\
\>", "Text",
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Number theory research expanded even more in the next century. Mathematicians \
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number theory and combinatorial number theory, complements to classic and \
analytic number theory. These new subfields developed the subject beyond \
anything Fermat could have dreamed of, with new abstract concepts and \
sophisticated techniques.\
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Later in the twentieth century, the invention of electronic computers gave \
number theorists the ability to answer questions faster. After this, number \
theory became a practical field. It was used to design encryption systems for \
governments and businesses. Number theory helped to factor very large numbers \
into primes, a process the code user knows and the code breakers do not.\
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Number theory is no longer considered as the purest of pure math. \
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The number theorist Leonard Dickson once said, \[OpenCurlyDoubleQuote]Thank \
God that number theory is unsullied by any application.\
\[CloseCurlyDoubleQuote] \
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But such a view is not true now. In 1974, Donald Knuth said, \
\[OpenCurlyDoubleQuote]\[Ellipsis]virtually every theorem in elementary \
number theory arises in a natural, motivated way in connection with the \
problem of making computers do high-speed numerical calculations.\
\[CloseCurlyDoubleQuote]\
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This lesson has already covered the main advantages of number theory in \
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Pythagoras\[CloseCurlyQuote]s dictum that \[OpenCurlyDoubleQuote]all is \
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Knowing how to solve problems that baffled even the Greeks, as well as the \
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