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Arithmetic progression. An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression.
[7] Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics". [8] However, though the Collatz conjecture itself remains open, efforts to solve the problem have led to new techniques and many partial results.
Sequences dn + a with odd d are often ignored because half the numbers are even and the other half is the same numbers as a sequence with 2d, if we start with n = 0. For example, 6n + 1 produces the same primes as 3n + 1, while 6n + 5 produces the same as 3n + 2 except for the only even prime 2. The following table lists several arithmetic ...
In number theory, primes in arithmetic progressionare any sequenceof at least three prime numbersthat are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by an=3+4n{\displaystyle a_{n}=3+4n}for 0≤n≤2{\displaystyle 0\leq n\leq 2}. According to the Green–Tao theorem, there ...
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset of integers and a target-sum , and the question is to decide whether any subset of the integers sum to precisely .[ 1 ] The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too ...
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory.Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.
Put your Halloween trivia knowledge to the test. Ranging from easy to hard, our questions cover Halloween-themed pop culture, history, and more fun facts.
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements. If two numbers by multiplying one another make somenumber, and any prime number measure the product, it willalso measure one of the original numbers. — Euclid, Elements Book VII, Proposition 30.
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