Rainbow numbers for x_1+x_2=kx_3 in ℤ_n

In this work, we investigate the fewest number of colors needed to guarantee a rainbow solution to the equation x_1 + x_2 = k x_3 in ℤ_n. This value is called the Rainbow number and is denoted by rb(ℤ_n, k) for positive integer values of n and k. We find that rb(ℤ_p, 1) = 4 for all primes greater than 3 and that rb(ℤ_n, 1) can be deterimined from the prime factorization of n. Furthermore, when k is prime, rb(ℤ_n, k) can be determined from the prime factorization of n.