AbstractLet θ(k, p) be the least s such that the congruence x1k + … + xsk ≡ 0(mod p) has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: θ(k) = O(k12+ϵ)
Let p be an odd prime. In 2008 E. Mortenson proved van Hamme’s following conjecture: (p−1)/2∑ k=
It is proved that the number of a ∈ {1, · · · , p − 1} which can be represented as a product of two ...
AbstractFor any prime p congruent to 1 modulo 4, let (t+up)/2 be the fundamental unit of Q(p). Then ...
AbstractLet θ(k, p) be the least s such that the congruence x1k + … + xsk ≡ 0(mod p) has a nontrivia...
AbstractLet θ(k, pn) be the least s such that the congruence x1k + ⋯ + xsk ≡ 0 (mod pn) has a nontri...
AbstractAn answer is given for a problem of Chowla and Shimura concerning congruences of the type a1...
AbstractLet In={1,2,…,n} and x:In↦R be a map such that ∑i∈Inxi⩾0. (For any i, its image is denoted b...
AbstractThe equation y2 ≡ x(x + a1)(x + a2) … (x + ar) (mod p), where a1, a2, …, ar are integers is ...
We consider the problem of describing all non-negative integer solutions to a linear congruence in m...
1997 We prove some congruences for the numbers N(m, n)=k ( nk) 2 ( n+kk) m. In particular, we show t...
Let p be a large prime number, K,L,M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of ou...
AbstractAs an extension of the Dirichlet divisor problem, S. Chowla and H. Walum conjectured that, a...
AbstractIn 1965, Chowla and Walum conjectured that, Ga,k(x):= Σn ≤ √x na Pk(xn) = O(xa2 + 14 + ε) ho...
AbstractLet k, l denote positive integers with (k, l) = 1. Denote by p(k, l) the least prime p ≡ l(m...
AbstractIt is proved that (for every ε > 0) ∑n⩽T13 ∑n<Tn12 namb Bk({Tnm}) = O(T(a+b+1)3ϵ) (where {·}...
Let p be an odd prime. In 2008 E. Mortenson proved van Hamme’s following conjecture: (p−1)/2∑ k=
It is proved that the number of a ∈ {1, · · · , p − 1} which can be represented as a product of two ...
AbstractFor any prime p congruent to 1 modulo 4, let (t+up)/2 be the fundamental unit of Q(p). Then ...
AbstractLet θ(k, p) be the least s such that the congruence x1k + … + xsk ≡ 0(mod p) has a nontrivia...
AbstractLet θ(k, pn) be the least s such that the congruence x1k + ⋯ + xsk ≡ 0 (mod pn) has a nontri...
AbstractAn answer is given for a problem of Chowla and Shimura concerning congruences of the type a1...
AbstractLet In={1,2,…,n} and x:In↦R be a map such that ∑i∈Inxi⩾0. (For any i, its image is denoted b...
AbstractThe equation y2 ≡ x(x + a1)(x + a2) … (x + ar) (mod p), where a1, a2, …, ar are integers is ...
We consider the problem of describing all non-negative integer solutions to a linear congruence in m...
1997 We prove some congruences for the numbers N(m, n)=k ( nk) 2 ( n+kk) m. In particular, we show t...
Let p be a large prime number, K,L,M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of ou...
AbstractAs an extension of the Dirichlet divisor problem, S. Chowla and H. Walum conjectured that, a...
AbstractIn 1965, Chowla and Walum conjectured that, Ga,k(x):= Σn ≤ √x na Pk(xn) = O(xa2 + 14 + ε) ho...
AbstractLet k, l denote positive integers with (k, l) = 1. Denote by p(k, l) the least prime p ≡ l(m...
AbstractIt is proved that (for every ε > 0) ∑n⩽T13 ∑n<Tn12 namb Bk({Tnm}) = O(T(a+b+1)3ϵ) (where {·}...
Let p be an odd prime. In 2008 E. Mortenson proved van Hamme’s following conjecture: (p−1)/2∑ k=
It is proved that the number of a ∈ {1, · · · , p − 1} which can be represented as a product of two ...
AbstractFor any prime p congruent to 1 modulo 4, let (t+up)/2 be the fundamental unit of Q(p). Then ...
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