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)
SummaryFor every t there is an explicitly given number k0 such that the equation 1k + 2k + + (x − 1)...
AbstractWe consider the problem of describing all non-negative integer solutions to a linear congrue...
AbstractFor any prime p congruent to 1 modulo 4, let (t+up)/2 be the fundamental unit of Q(p). Then ...
AbstractLet θ(k, pn) be the least s such that the congruence x1k + ⋯ + xsk ≡ 0 (mod pn) has a nontri...
AbstractLet θ(k, p) be the least s such that the congruence x1k + … + xsk ≡ 0(mod p) has a nontrivia...
AbstractAn answer is given for a problem of Chowla and Shimura concerning congruences of the type a1...
AbstractThe equation y2 ≡ x(x + a1)(x + a2) … (x + ar) (mod p), where a1, a2, …, ar are integers is ...
AbstractIn 1965, Chowla and Walum conjectured that, Ga,k(x):= Σn ≤ √x na Pk(xn) = O(xa2 + 14 + ε) ho...
We obtain a lower bound for the minimum over positive integers such that the sum of certain powers o...
AbstractIn 1979 R. Apéry introduced the numbers an = Σ0n(kn)2(kn + k) and un = Σ0n(kn)2(kn + k)2 in ...
AbstractIf lr(p) is the least positive integral value of x for which y2 ≡ x(x + 1) ⋯ (x + r − 1)(mod...
We study the set D of positive integers d for which the equation $\phi(a)-\phi(b)=d$ has infinitely ...
Let K be a number field, S be the set of primes of K above 2 and T the subset of primes above 2 havi...
AbstractWe define h(n) to be the largest function of n such that from any set of n nonzero integers,...
In this paper, we study various arithmetic properties of the function p¯¯¯2,k(n), which denotes the ...
SummaryFor every t there is an explicitly given number k0 such that the equation 1k + 2k + + (x − 1)...
AbstractWe consider the problem of describing all non-negative integer solutions to a linear congrue...
AbstractFor any prime p congruent to 1 modulo 4, let (t+up)/2 be the fundamental unit of Q(p). Then ...
AbstractLet θ(k, pn) be the least s such that the congruence x1k + ⋯ + xsk ≡ 0 (mod pn) has a nontri...
AbstractLet θ(k, p) be the least s such that the congruence x1k + … + xsk ≡ 0(mod p) has a nontrivia...
AbstractAn answer is given for a problem of Chowla and Shimura concerning congruences of the type a1...
AbstractThe equation y2 ≡ x(x + a1)(x + a2) … (x + ar) (mod p), where a1, a2, …, ar are integers is ...
AbstractIn 1965, Chowla and Walum conjectured that, Ga,k(x):= Σn ≤ √x na Pk(xn) = O(xa2 + 14 + ε) ho...
We obtain a lower bound for the minimum over positive integers such that the sum of certain powers o...
AbstractIn 1979 R. Apéry introduced the numbers an = Σ0n(kn)2(kn + k) and un = Σ0n(kn)2(kn + k)2 in ...
AbstractIf lr(p) is the least positive integral value of x for which y2 ≡ x(x + 1) ⋯ (x + r − 1)(mod...
We study the set D of positive integers d for which the equation $\phi(a)-\phi(b)=d$ has infinitely ...
Let K be a number field, S be the set of primes of K above 2 and T the subset of primes above 2 havi...
AbstractWe define h(n) to be the largest function of n such that from any set of n nonzero integers,...
In this paper, we study various arithmetic properties of the function p¯¯¯2,k(n), which denotes the ...
SummaryFor every t there is an explicitly given number k0 such that the equation 1k + 2k + + (x − 1)...
AbstractWe consider the problem of describing all non-negative integer solutions to a linear congrue...
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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