Abstract
M. Pohst asked the following question: is it true that every prime can be written in the form 2u ± 3v with some non-negative integers u, v? We put the problem into a general framework, and prove that the length of any arithmetic progression in t-term linear combinations of elements from a multiplicative group of rank r (e.g. of S-units) is bounded in terms of r, t, n, where n is the number of the coefficient t-tuples of the linear combinations. Combining this result with a recent theorem of Green and Tao on arithmetic progressions of primes, we give a negative answer to the problem of M. Pohst.
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Communicated by Attila Pethő
Research supported in part by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and by the OTKA grants T042985 and T048791.
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Hajdu, L. Arithmetic Progressions in Linear Combinations of S-Units. Period Math Hung 54, 175–181 (2007). https://doi.org/10.1007/s-10998-007-2175-8
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DOI: https://doi.org/10.1007/s-10998-007-2175-8