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Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8.
If k is an odd integer, then 3k + 1 is even, so 3k + 1 = 2 a k ′ with k ′ odd and a ≥ 1. The Syracuse function is the function f from the set I of positive odd integers into itself, for which f(k) = k ′ (sequence A075677 in the OEIS). Some properties of the Syracuse function are: For all k ∈ I, f(4k + 1) = f(k). (Because 3(4k + 1) + 1 ...
McCullough and Wade [18] extended this approach, which produces all Pythagorean triples when k > h √ 2 /d: Write a positive integer h as pq 2 with p square-free and q positive. Set d = 2pq if p is odd, or d= pq if p is even. For all pairs (h,k) of positive integers, the triples are given by
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers. [1] It is closely related to but stronger than Legendre's conjecture, Andrica's conjecture, and Brocard's conjecture. It is named after Danish mathematician Ludvig Oppermann, who announced it in an unpublished lecture in March 1877. [2]
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. [1] [2] The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.
Disregarding the above definition of n!! for even values of n, the double factorial for odd integers can be extended to most real and complex numbers z by noting that when z is a positive odd integer then [18] [19]
In mathematics, an interprime is the average of two consecutive odd primes. [1] For example, 9 is an interprime because it is the average of 7 and 11. The first interprimes are: