Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
نویسنده
چکیده
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 this lower bound by an additional factor of r for all α, r ≥ 3.
منابع مشابه
Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
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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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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