Resumen de Repeating Decimals: A Period Piece

Kenneth Ross

• Consider the repeating decimal of a reduced fraction t/n between 0 and 1, where none of the primes 2, 3 or 5 is a factor of n. The fraction satisfies the m-block property if m divides the period ? and, when the repeating portion is broken into m blocks of equal length, the sum of the m blocks is a string of nines or an integer multiple of a string of nines. An easily-verified sufficient condition is given that implies t/n has the m-block property. The condition is not necessary. For n equal to a prime power pa (p >= 7), the condition is shown to be sufficient as well. Moreover, t/pa has the m-block property if and only if m is not a power of p. (These general results for prime powers seem to be new.) The 2-block property for primes is known as Midy�s theorem (1836). It holds for all prime powers, but not in general. Many examples and a brief history are included.