What does number theory study?

It studies integers and their properties and relationships, including primes, divisibility, and objects or equations connected with integers.

What are the integers?

The integers are the whole numbers extending in both directions, including negative numbers, zero, and positive numbers.

How does number theory differ from arithmetic?

Arithmetic broadly studies numerical operations and may include real numbers; number theory concentrates on integer properties and relationships.

What is a Diophantine equation?

It is an equation for which integer or rational solutions are sought.

Why is the RSA cryptosystem a number theory application?

RSA uses properties of primes and the computational difficulty of factoring large composite integers to support public-key cryptography.

Discrete mathematicsLesson 1 of 1 in the course Number theory and recurrences

Number Theory: Integer Structure, Methods, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Number theory studies integers and the patterns, operations, and relationships associated with them. This lesson introduces the subject, explains how it differs from general arithmetic, and traces important landmarks in its history.

The integers are the whole numbers, including positive numbers, zero, and negative numbers: $\ldots,-3,-2,-1,0,1,2,3,\ldots$ . Number theory focuses on their properties and relationships. It also studies objects built from integers, such as rational numbers, and generalizations such as algebraic integers.

All flashcards

What does number theory study?: It studies integers and their properties and relationships, including primes, divisibility, and objects or equations connected with integers.
What are the integers?: The integers are the whole numbers extending in both directions, including negative numbers, zero, and positive numbers.
How does number theory differ from arithmetic?: Arithmetic broadly studies numerical operations and may include real numbers; number theory concentrates on integer properties and relationships.
What is a Diophantine equation?: It is an equation for which integer or rational solutions are sought.
Why is the RSA cryptosystem a number theory application?: RSA uses properties of primes and the computational difficulty of factoring large composite integers to support public-key cryptography.

Quiz questions

For a positive integer $n$ , when are integers $a$ and $b$ congruent modulo $n$ ?Answer: When $n$ divides $a-b$ , so they have the same remainder.
What does Fermat's little theorem state when $p$ is prime and $p$ does not divide $a$ ?Answer: $a^{p-1}\equiv1\pmod p$
What kind of problem does the Chinese remainder theorem help solve?Answer: Finding integers that satisfy several congruences, especially with pairwise coprime moduli.
What does the prime-counting function $\pi(x)$ count?Answer: The number of primes less than or equal to $x$ .
Which description defines an algebraic number?Answer: A complex root of some nonzero polynomial with rational coefficients.

Based on the Wikipedia article Number theory, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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