“prime”的概念、定义、翻译、参考文献-科学参考

prime

Mathematics

A positive integer p is a prime if p ≠ 1 and its only positive divisors are 1 and itself.

It is known that there are infinitely many primes. Euclid’s proof argues by contradiction as follows. Suppose that there are finitely many primes p₁, p₂,…,p_n. Consider the number p₁p₂×… ×p_n + 1. This is not divisible by any of p₁, p₂,…, p_n—division by each leaves a remainder of 1—so it is either another prime itself or is divisible by primes not among the p_i. It follows that the number of primes is not finite.

The notion of a prime number generalizes to that of a prime element in the theory of rings. At any time, the largest known prime is usually the largest known Mersenne prime. There are many deep results in number theory relating to prime numbers, such as Fermat’s Two Square Theorem, Bertrand’s postulate, and the Green-Tao theorem, and many important open problems, such as Goldbach’s conjecture, Legendre’s conjecture, Riemann hypothesis, and Twin prime conjecture.

See also Fundamental Theorem of Arithmetic, prime number theorem.

Logic

1. In proof theory, any formula that is either atomic or a propositional constant (e.g., verum ( $⊤$ )).
2. Describes any theory $T$ such that for a disjunction connective $\lor$ and formulae $φ$ and $ψ$ in the language of $T$ , the following holds:
$φ \lor ψ \in T$ if and only if ${\begin{matrix} φ \in T, or \\ ψ \in T \end{matrix}$

单词	prime
释义	prime Mathematics A positive integer p is a prime if p ≠ 1 and its only positive divisors are 1 and itself. It is known that there are infinitely many primes. Euclid’s proof argues by contradiction as follows. Suppose that there are finitely many primes p₁, p₂,…,p_n. Consider the number p₁p₂×… ×p_n + 1. This is not divisible by any of p₁, p₂,…, p_n—division by each leaves a remainder of 1—so it is either another prime itself or is divisible by primes not among the p_i. It follows that the number of primes is not finite. The notion of a prime number generalizes to that of a prime element in the theory of rings. At any time, the largest known prime is usually the largest known Mersenne prime. There are many deep results in number theory relating to prime numbers, such as Fermat’s Two Square Theorem, Bertrand’s postulate, and the Green-Tao theorem, and many important open problems, such as Goldbach’s conjecture, Legendre’s conjecture, Riemann hypothesis, and Twin prime conjecture. See also Fundamental Theorem of Arithmetic, prime number theorem. Logic 1. In proof theory, any formula that is either atomic or a propositional constant (e.g., verum ( $⊤$ )). 2. Describes any theory $T$ such that for a disjunction connective $\lor$ and formulae $φ$ and $ψ$ in the language of $T$ , the following holds: $φ \lor ψ \in T$ if and only if ${\begin{matrix} φ \in T, or \\ ψ \in T \end{matrix}$
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