- If n is 2 and p is a prime number, then only half
the numbers from 1 to p-1 appear in the bottom row and each
number that does appear, appears twice. Hence, half the numbers
have square roots and half have no square roots.
- If n is 2 and p is a prime number, the numbers in
the bottom row form a palindrome (hence square roots are +/- some
- If n is 2 and p is a prime number then 1 shows up
only at the extreme left and extreme right of the bottom row. This
means the square root of 1 is only +1 mod p and -1
mod p. If p is not a prime, then 1 may show up
elsewhere in the bottom row. But this is not always the case. For
example, if p is 6,8,9,10,12,14,15,16,18,20,21 then 1
doesn't, does, doesn't, doesn't, does, doesn't, does, does, doesn't,
does, does show up elsewhere in the bottom row, repectively. Hence,
1 mod p has only trivial roots of unity if p is a
prime number but about half the time has non trivial roots
if p is not prime.
- If p is a prime number and n is p-1 then
all the numbers in the bottom row are 1. This is a demonstration
of Fermat's Little Theorem. For example, set p to 19 and
n to 18. Try a non-prime for p, such as 14, and try
all values of n from 1 to p. For p non-prime,
and any n less than p, why must some number in the
bottom row be different from 1?