The spread polynomial, denoted by , is recursively defined by , , and
for . We will, for the context of this note, restrict our discussion to the finite field , where is an odd prime. As an example, we see that in ,
In fact, for any positive integer we will have that . Thus, a natural question to be asked is whether this phenomenon is also seen for arbitrary finite fields; as it turns out, it does and in fact we have in , where . How do we prove that this is true? Are we able to discern other patterns within particular values of ? If so, how do we ascertain this mathematically?