Midy's Theorem for Periodic Decimals

Abstract

The decimal expansion of 1/7 is 0.142857142857..., the block 142857 repeating forever. We call 142857 the period and its length is 6 = 2x3. If the period is broken into 2 pieces each of length 3 which are then added, the result is 142 + 857 = 999; similarly 14 + 28 + 57 = 99. Other periodic decimals show the same phenomenon while others do not. The general question then arises: Let a/N be a fraction with denominator prime to 10, having decimal expansion with period length dk, if the period is broken into d pieces of length k which are then added, when will the sum be 99...9, a block of k 9's? In 1836 E. Midy published an article analyzing this question and gave sufficient conditions in certain cases. We put the problem in a more general setting, working in an arbitrary number base B, and obtain new results. However, some open questions still remain.