ABSTRACT
In Chapter 3, we study the number of solutions of the congruence (cong form) when the greatest common divisors (gcds) of coordinates x i and the modulus n are certain numbers. More formally, we study solutions when (x i,n) = t i (1≤ i ≤ k), where t 1, …, t k are given positive divisors of n. To see how general the problem is, note that when all gcds are equal to one (that is, when ti = 1) then the problem is asking for the solutions of the congruence (cong form) when xi ∈ Z n *, where Z n * is the multiplicative group of integers modulo n. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions. Some of these applications are discussed in the next chapters.
