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(also **arithmetic series**)

1A sequence of numbers in which each differs from the preceding by a constant quantity (e.g., 3, 6, 9, 12, etc.; 9, 7, 5, 3, etc.).

*‘Dutch mathematician Bartel L. van der Waerden was for one of the first to identify such a pattern among integers - one that involves arithmetic progressions in sets of numbers.’**‘The following year he published An elementary proof of the prime number theorem for arithmetic progressions.’**‘In one step toward elucidating certain primal mysteries, two mathematicians have now apparently proved that the population of primes contains an infinite collection of arithmetic progressions.’**‘This work on polygonal numbers is related to the ideas on arithmetic progressions that appear in another work by Hypsicles, making it more likely that indeed Hypsicles had indeed done original work on this topic.’**‘There are problems on the least common multiple and arithmetic progressions.’**‘Bombieri applied his improved large sieve method to prove what is now called ‘Bombieri's mean value theorem ‘, which concerns the distribution of primes in arithmetic progressions.’’**‘For example, he discussed some problems concerning arithmetic progressions.’**‘In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.’**‘Analytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet's memoir of 1837 on the existence of primes in a given arithmetic progression.’**‘Its 92 problems illustrate the formula for summing an arithmetic progression.’*

- 1.1The relationship between numbers in an arithmetic progression.
*‘the numbers are in arithmetic progression’**‘Dubner, Zimmermann, and Forbes are now looking for help to find a sequence of nine consecutive primes in arithmetic progression.’**‘In this instance, the integers 6, 9, 1 constitute a three-term rainbow arithmetic progression with a common difference of 3.’**‘A theorem of Peter Gustav Lejeune Dirichlet on primes in arithmetic progression guarantees that all the other notes are heard infinitely often when one plays all the primes.’**‘As a bonus, the project participants also found 27 new sets of eight consecutive primes in arithmetic progression and several hundred sets of seven primes.’**‘Whereas in Carnatic rhythms, tempos ascend in geometric progression, the corresponding changes in Kerala rhythms occur in arithmetic progression.’**‘They do not have to be in arithmetic progression as in the above examples: 2-6 - 8 would be OK.’*

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