For some prime numbers p, the equation x x 1 D 0 has a solution in Zp. For other same as

Chapter 37, Problem 37.5

(choose chapter or problem)

For some prime numbers p, the equation x x 1 D 0 has a solution in Zp. For other same as in ordinary arithmetic. The expression x x 1 should be parenthesized as .x x/ 1. In essence, this problem is asking you to determine whether or not there is a p 1 in Zp for various prime numbers p. primes it does not. For example, in Z17 we have 4 4 1 D 0, but in Z19 there is no solution. The equation has a solution for p D 2, but this is not a particularly interesting example. Investigate the first several (say, to 103) odd prime numbers p and divide them into two categories: those for which x x 1 D 0 has a solution in Zp and those for which it does not. We recommend that you write a computer program to do this. State a conjecture based on your evidence.