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Consider the linear congruence equation for , . Let denote the generalized gcd of and which is the largest with dividing and simultaneously. Let be all positive divisors of . For each , define . K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on . We generalize their result with generalized gcd restrictions on and prove that for the above linear congruence, the number of solutions...
For a prime and positive integers with , we show that , the number of simultaneous solutions in to , , , satisfiesWhen we obtain a precise asymptotic count on . This leads to the new twisted exponential sum boundfor trinomials , and to results on the average size of such sums.
We shall describe how to construct a fundamental solution for the Pell equation over finite fields of characteristic . Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation .
For every finite Abelian group Γ and for all , if there exists a solution of the equation in non-negative integers , where are positive integers, then the number of such solutions is estimated from below in the best possible way.
The number of solutions of the congruence in the box is estimated from below in the best possible way, provided for all i,j either or or .
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