12/11/2023 0 Comments 4.2 constructing arithmetic sequences![]() ![]() NOTE: This is not my area of research so this question is mostly out of curiosity. So my question is what are the known techniques for constructing a sequence of primes of length $k$? How would one find the "first" prime in the sequence or even the "largest prime" that would satisfy the sequence (assuming there is one)? Also, while the theorem gives a lower bound for $d$, is it known if it is the sharpest lowest bound there is? 2 4.1 Identifying Graphing Sequences 161-163 1-20 20-Oct-17 23-Oct-17 3 4.2 Constructing Arithmetic Sequences 170-173 1-21 23-Oct-17 24-Oct-17 4 4. Using this we can get that the prime $p=5$ and $d = 6$ will result in a sequence primes in arithmetic progression of length $5$. So for instance if you want a sequence of primes in arithmetic progression of length $5$ ie If all the terms of the arithmetic progressionĪre prime numbers then the common difference $d$ is divisible by every prime $q ![]() If you would like to learn more, I suggest reading Julia Wolf's excellent survey article Arithmetic and polynomial progressions in the primes, d'après Gowers, Green, Tao and Ziegler. (It has not yet been published) This means that we unconditionally have asymptotics for the number of primes in a $k$ term arithmetic progression. Recently, Green, Tao and Ziegler resolved the Gowers inverse conjecture, and their paper is currently on the arxiv. The 3rd term minus the 2nd term (11 9) is 2. Consider this sequence, defined by the explicit rule +5 Domain Range The 2nd term minus the 1st term (9 7) is 2. This difference, written as d, is called the common difference. We see that the ratio of any term to the preceding term is 1 3. In an arithmetic sequence, the difference between consecutive terms is always equal. In a geometric sequence, the ratio of every pair of consecutive terms is the same. In a paper published in the Annals in 2012, Green and Tao resolve the MN conjecture, proving that the Möbius Function is strongly orthogonal to nilsequences. In general, an arithmetic sequence is any sequence of the form an cn + b. They assumed the Möbius Nilsquence conjecture (MN) and the Gowers Inverse norm conjecture (GI). The primorial $n\#=\prod_\right).$$Äo note however, that Green and Tao's paper made two major assumptions. It's conjectured that there is an AP of length $n$ starting with the smallest prime $\ge n$. This records page seems to be a good reference.
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