An Introduction to the Theory of Numbers

Hardy, G. H., E. M. Wright, A. I. t. t. T. o. N., & J. H. Silverman, O. U. P. (2008). An Introduction to the Theory of Numbers. Oxford University Press.

Type

Book chapter

Intellectual base

Textbook

Year

2008

ISBN

9780199219865

Link

https://global.oup.com/academic/product/an-introduction-to-the-theory-of-numbers-9780199219865

Cited from

mathematics

Cited by

11 citations across 11 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Factorization
  • In computer science it is code refactoring, database normalization, and the Chinese Remainder Theorem enabling computation across coprime moduli.

    This sourceStandard reference for prime factorization, the fundamental theorem of arithmetic, and the Chinese Remainder Theorem enabling computation across coprime moduli.

• Predicate
  • The sharp boundary is total: every natural number is unambiguously prime or not, with no third value. Negation and Boolean closure are immediately available: \(\neg P(n)\) ("is composite or one") is itself a predicate whose extension is the complement of \(S\), and conjunction lets us build \(\{\, n : P(n) \wedge n \equiv 1 \pmod 4 \,\}\) as a set operation on extensions. Quantifiers turn the predicate into propositions: \(\forall n\, (n > 2 \wedge P(n) \to n \text{ is odd})\) is a true universal claim about the satisfier-set, while the twin-prime conjecture is an open existential one.

    This sourceStandard number-theory reference, including primality and open conjectures such as the twin-prime conjecture.

• Proof By Contradiction
  • In mathematics the canonical cases are Euclid's proof of the infinitude of primes, the irrationality of the square root of two, and Cantor's diagonal argument; whole branches of analysis and topology rest on contradiction-style reasoning.

    This sourceStandard reference for Euclid's proof of the infinitude of primes and the irrationality of √2 by contradiction.

Domain-specific

• Additive function
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Deficient number
  • A positive integer \(n\) is deficient when \(s(n)<n\), where \(s(n)\) is the sum of positive proper divisors; since \(s(n)=\sigma(n)-n\), this is equivalent to \(\sigma(n)<2n\).
• Division (mathematics)
  • Division is the inverse problem for multiplication: determine q such that bq=a when possible; in Euclidean division determine unique q,r with a=bq+r under a remainder bound.
• Perfect number
  • A perfect number is a positive integer \(n\) satisfying \(s(n)=n\), where \(s(n)\) sums the positive proper divisors of \(n\); equivalently, \(\sigma(n)=2n\).
• Reduced residue system
  • For example, the distinction between constitutive identity and a convenient observable transfers from Modulo 8, the set {1,3,5,7} is a reduced residue system. to Multiplying a reduced residue system by a unit modulo n permutes its residue classes..
• Square-Free Element

  • This sourceStandard source for square-free integers, prime-factor support, the Möbius function, and density; used only for the integer instance.

• Square number
  • … directly demonstrates that the integer domain, inclusion of zero, witness integer or exact square root, exponentiation convention, sign and any modular or factorization criterion are explicit. to An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation..
• Sum of squares function
  • The move from For \(k=2\), \(r_2(1)=4\), represented by \((\pm1,0)\) and \((0,\pm1)\). to Jacobi's four-square formula gives \(r_4(n)=8\sum_{d\mid n,\,4\nmid d}d\). demonstrates that continuity.

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