Problem. If p be one of the roots of the equation xm − 1 = 0, (not 1,) then (1 − p)(1 − p2) …. (1 − pm − 1) = m, provided m is a prime number.

If m be not a prime number, and if , the same will hold for all roots p = p1α, where α is a number < m and prime to m. But for all roots p = p1α, where α, or one of its prime factors, is also a prime factor of m, the product (1 − p)(1 − p2) …. (1 − pm − 1) will be equal to 0.