Document Type
Lecture
Publication Date
3-19-2019
Abstract
In this talk, we will consider the congruence equation a + b ≡ c (mod p) with 1 ≤ a, b, c ≤ H where x stands for the multiplicative inverse of x (mod p). We prove that its number of solutions is asymptotic to H³/p when H > p²/³+o(1) by estimating a certain average of Kloosterman sums via Gauss sums. On the other hand, when H < p¹/²√log p, the number of solutions has order of magnitude H log H. It would be interesting to understand better its transition of behavior. By transforming the question slightly, one can relate the problem to a certain first moment of Dirichlet L-functions at s = 1. This is still work in progress.
Relational Format
presentation
Recommended Citation
Chan, Tsz Ho, "On the Congruence Equation a + b ≡ c (mod p)" (2019). Algebra/Number Theory Seminar. 26.
https://egrove.olemiss.edu/math_algebra_number_theory/26
