Enumeration Problems for a Linear Congruence Equation


The Taiwanese Journal of Mathematics, is published by the Mathematical Society of the Republic of China, http://journal.taiwanmathsoc.org.tw/index.php/TJM/index.


Let m ≥ 2 and r ≥ 1 be integers and let c Є Zm = {0, 1, …,m ─ 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, …, xn Є Zm of the congruence x1 + x2 + ••• + xr ≡ c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has interrelationship with the restricted integer partition.