_{n }under exponentiation where n is the product of two primes. Each entry in each exponentiation table here is constructed by raising the row index a to the power of the column index b mod n. Note that since this construction is not Abelian the opposite statement does not hold in its exponentiation tables. The post was inspired by a small discovery I came across while taking CMPUT 210 at the University of Alberta.

Z

_{n}is closed with respect to exponentiation since for all a, b in Z

_{n}, a

^{b}= a*a*a*a*... (b factors) and since Z

_{n}is closed with respect to multiplication then a*a*a*a*... (b factors) is also in Z

_{n}.

However, Z

_{n}under exponentiation is not Abelian since a

^{b}is not always congruent to b

^{a}mod n, therefore most of what was said about multiplication tables in Z sub n and addition tables in Z sub n doesn't necessarily hold.

In fact, Z

_{n}under exponentiation as constructed below where n is the product of two primes is not even a group.

**So, is it still possible to generate exponentiation tables for Z**

_{n }without actually computing the greatest common divisor of (a^{b}, n) for all a and b in Z_{n}?