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A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet). The number n can be any nonnegative integer . For example, a complex number can be represented as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as ...
An equivalent definition of Skewes' number exists for prime k-tuples (Tóth (2019)). ... denote a prime (k + 1)-tuple, () the number of primes below ...
In number theory, the first Hardy–Littlewood conjecture [1] states the asymptotic formula for the number of prime k-tuples less than a given magnitude by generalizing the prime number theorem. It was first proposed by G. H. Hardy and John Edensor Littlewood in 1923. [2]
In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {}. In axiomatic set theory , the existence of singletons is a consequence of the axiom of pairing : for any set A , the axiom applied to A and A asserts the existence of { A , A } , {\displaystyle \{A,A\},} which is the same as the ...
The left and right projection of a pair p is usually denoted by π 1 (p) and π 2 (p), or by π ℓ (p) and π r (p), respectively. In contexts where arbitrary n-tuples are considered, π n i (t) is a common notation for the i-th component of an n-tuple t.
Total number of puzzles solved this year: 11.1 billion. The Mini Crossword Plays: 2.1 billion. Strands Plays: 1.3 billion. ... The 1,000th Wordle puzzle ran in March, a milestone for the popular ...
More generally, a positive integer c is the hypotenuse of a primitive Pythagorean triple if and only if each prime factor of c is congruent to 1 modulo 4; that is, each prime factor has the form 4n + 1. In this case, the number of primitive Pythagorean triples (a, b, c) with a < b is 2 k−1, where k is the number of distinct prime factors of c ...
The best ideas for things to do on New Year's Eve 2024, including fun ways to celebrate at home and inspiring New Year's activities for any age or group size.