Nontrivial upper bounds for the least common multiple of an arithmetic progression
نویسندگان
چکیده
منابع مشابه
Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
For relatively prime positive integers u0 and r, we consider the arithmetic progression {uk := u0 + kr} n k=0 . We obtain a new lower bound on Ln := lcm{u0, u1, . . . , un}, the least common multiple of the sequence {uk} n k=0 . In particular, we show that Ln ≥ u0r(r + 1) whenever α ≥ 1 and n ≥ 2αr; this result improves the best previous bound for all but three choices of α, r ≥ 2. We sharpen t...
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For relatively prime positive integers u0 and r, we consider the arithmetic progression {uk := u0 + kr} n k=0 . Define Ln := lcm{u0, u1, . . . , un} and let a ≥ 2 be any integer. In this paper, we show that, for integers α, r ≥ a and n ≥ 2αr, we have Ln ≥ u0r (r + 1). In particular, letting a = 2 yields an improvement to the best previous lower bound on Ln (obtained by Hong and Yang) for all bu...
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For relatively prime positive integers u0 and r, we consider the least common multiple Ln := lcm(u0, u1, . . . , un) of the finite arithmetic progression {uk := u0 + kr}k=0. We derive new lower bounds on Ln which improve upon those obtained previously when either u0 or n is large. When r is prime, our best bound is sharp up to a factor of n + 1 for u0 properly chosen, and is also nearly sharp a...
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We settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.
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Let nr(a,q) be the least /--free number in the arithmetic progession a modulo q. Several results are proved that give lower bounds for n,.(a, q), improving on previous results due to Erdös and Warlimont. In addition, a heuristic argument is given, leading to two conjectures that would imply that the results of the paper are close to best possible.
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ژورنال
عنوان ژورنال: Asian-European Journal of Mathematics
سال: 2020
ISSN: 1793-5571,1793-7183
DOI: 10.1142/s1793557121501382