AbstractLetG(k, r) denote the smallest positive integergsuch that if 1=a1, a2, …, agis a strictly increasing sequence of integers with bounded gapsaj+1−aj⩽r, 1⩽j⩽g−1, then {a1, a2, …, ag} contains ak-term arithmetic progression. It is shown thatG(k, 2)>(k−1)/2(43)(k−1)/2,G(k, 3)>(2k−2/ek)(1+o(1)),G(k, 2r−1)>(rk−2/ek)(1+o(1)),r⩾2
AbstractA classic theorem of van der Waerden asserts that for any positive integer k, there is an in...
We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denu...
AbstractIn this paper we obtain an improved asymptotic formula on the frequency of k-free numbers wi...
AbstractLetG(k, r) denote the smallest positive integergsuch that if 1=a1, a2, …, agis a strictly in...
AbstractFork>2 andr⩾2, letG(k,r) denote the smallest positive integergsuch that every increasing seq...
AbstractDenote by B(k, l) the least integer such that, if the numbers 1, 2, 3,…, B(k, l) + 1 are par...
Let f_(s, k)(n) be the maximum possible number of s‐term arithmetic progressions in a set of n integ...
AbstractK. F. Roth (1964, Acta. Arith.9, 257–260) proved that the discrepancy of arithmetic progress...
AbstractFor given n, k, the minimum cardinal of any subset B of [1, n] which meets all of the k-term...
AbstractA new expansion is given for partial sums of Eulerʼs pentagonal number series. As a corollar...
AbstractWinkler has proved that, if n and m are positive integers with n ≤ m ≤ n25 and m ≡ n (mod 2)...
AbstractEstimating the discrepancy of the set of all arithmetic progressions in the first N natural ...
AbstractIt is known that the sequence 1,2,1,1,2,2,2,1,1,2,1,1,2,1,1,2,2,… of lengths of blocks of id...
AbstractIn this paper we prove the following result. Let p be a prime, and k a positive integer. Let...
AbstractWe prove in a strong form an old conjecture of Erdös to the effect that ∑1⩽i<j⩽T(n)(dj−di)−1...
AbstractA classic theorem of van der Waerden asserts that for any positive integer k, there is an in...
We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denu...
AbstractIn this paper we obtain an improved asymptotic formula on the frequency of k-free numbers wi...
AbstractLetG(k, r) denote the smallest positive integergsuch that if 1=a1, a2, …, agis a strictly in...
AbstractFork>2 andr⩾2, letG(k,r) denote the smallest positive integergsuch that every increasing seq...
AbstractDenote by B(k, l) the least integer such that, if the numbers 1, 2, 3,…, B(k, l) + 1 are par...
Let f_(s, k)(n) be the maximum possible number of s‐term arithmetic progressions in a set of n integ...
AbstractK. F. Roth (1964, Acta. Arith.9, 257–260) proved that the discrepancy of arithmetic progress...
AbstractFor given n, k, the minimum cardinal of any subset B of [1, n] which meets all of the k-term...
AbstractA new expansion is given for partial sums of Eulerʼs pentagonal number series. As a corollar...
AbstractWinkler has proved that, if n and m are positive integers with n ≤ m ≤ n25 and m ≡ n (mod 2)...
AbstractEstimating the discrepancy of the set of all arithmetic progressions in the first N natural ...
AbstractIt is known that the sequence 1,2,1,1,2,2,2,1,1,2,1,1,2,1,1,2,2,… of lengths of blocks of id...
AbstractIn this paper we prove the following result. Let p be a prime, and k a positive integer. Let...
AbstractWe prove in a strong form an old conjecture of Erdös to the effect that ∑1⩽i<j⩽T(n)(dj−di)−1...
AbstractA classic theorem of van der Waerden asserts that for any positive integer k, there is an in...
We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denu...
AbstractIn this paper we obtain an improved asymptotic formula on the frequency of k-free numbers wi...
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