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A system of simultaneous congruences arising from trinomial exponential sums

Todd Cochrane, Jeremy Coffelt, Christopher Pinner (2006)

Journal de Théorie des Nombres de Bordeaux

For a prime p and positive integers < k < h < p with d = ( h , k , , p - 1 ) , we show that M , the number of simultaneous solutions x , y , z , w in p * to x h + y h = z h + w h , x k + y k = z k + w k , x + y = z + w , satisfies M 3 d 2 ( p - 1 ) 2 + 25 h k ( p - 1 ) . When h k = o ( p d 2 ) we obtain a precise asymptotic count on M . This leads to the new twisted exponential sum bound x = 1 p - 1 χ ( x ) e 2 π i f ( x ) / p 3 1 4 d 1 2 p 7 8 + 5 h k 1 4 p 5 8 , for trinomials f = a x h + b x k + c x , and to results on the average size of such sums.

Number of solutions in a box of a linear equation in an Abelian group

Maciej Zakarczemny (2016)

Colloquium Mathematicae

For every finite Abelian group Γ and for all g , a , . . . , a k Γ , if there exists a solution of the equation i = 1 k a i x i = g in non-negative integers x i b i , where b i are positive integers, then the number of such solutions is estimated from below in the best possible way.

On a linear homogeneous congruence

A. Schinzel, M. Zakarczemny (2006)

Colloquium Mathematicae

The number of solutions of the congruence a x + + a k x k 0 ( m o d n ) in the box 0 x i b i is estimated from below in the best possible way, provided for all i,j either ( a i , n ) | ( a j , n ) or ( a j , n ) | ( a i , n ) or n | [ a i , a j ] .

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Given an integer n 3 , let u 1 , ... , u n be pairwise coprime integers 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 { ± 1 } . Does there exist at least one prime q such that q divides i I u i - ε ( I ) for some I 𝒟 , but it does not divide u 1 u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions.We use the result to prove that, if ε 0 { ± 1 } and A is a set of three or more primes that contains all prime divisors of any number of the form p B p - ε 0 for...

On Equations y² = xⁿ+k in a Finite Field

A. Schinzel, M. Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Solutions of the equations y² = xⁿ+k (n = 3,4) in a finite field are given almost explicitly in terms of k.

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