If m, n are natural numbers, does the sequence m, m+n, m+2n, ... contain prime numbers? How many?
Elaborate on history, relation to other prime number problems, solutions.
It has long been considered that there are arbitrarily long arithmetic progressions of primes. For instance the longest known arithmetic sequence of primes is given by 43142746595714191+23681770*�223092870n for n = 0,1, ... ,25. Throughout history, Many mathematicians have worked for years trying to prove or disprove this conjecture, and more so to come up with a particular algorithm to find arithmetic progressions of primes of any given length. This conjecture has created many questions and problems that to this day, remain unanswered. If m, n are natural numbers, does the sequence m, m+n, m+2n, ... contain prime numbers? How many? Finite or infinite? Lengths of progressions?
Starting as early as 1770, Lagrange and Waring questioned and researched how large the common difference of an arithmetic progression of n primes must be. In 1923, Hardy and Littlewood made a came up with a conjecture that there are infinitely prime n-tuples of the form p, p + a1, .... ,p + an for any positive integers a1, ..... ,an. This idea was then adapted to the case where a1, ... ,an is a geometric sequence in the form a1, 2*a1, 3*a1, .... ,n*a1. Ever since, there has been a plethora of questions (some answered and some not) regarding primes in arithmetic progressions.
Test complete.
