Reptends and Reciprocals

Summary

Reptends and Reciprocals by Greg Egan Is there an integer d whose reciprocal, 1/d, has a decimal representation with a recurring block of digits exactly 10 digits long, containing all ten decimal digits? The answer is yes! 72,728 has this property. In fact, we can find integers with analogous properties in several different bases besides base 10. A number is described as pandigital in base b if its representation contains all the digits 0, ..., b – 1, and non-redundant if each digit appears only once, so we will call these blocks of b repeating digits, enclosed in parentheses in the table below, non-redundant pandigital reptends. Smallest integers whose reciprocals have non-redundant pandigital reptends Base 21/3=0.(01)2 ... Base 41/34=0.0(0132)4 ... Base 61/93=0.0(021534)6 ... Base 101/72,728=0.000(0137498625) ... Base 121/12,560=0.00(01798A2654B3)12 ... Base 141/28,784,914,432=0.000000000A09(7DAC59B6031842)14 ... Base 181/82,703,547,776=0.0000000(02731C6D8HFAEG5B49)18 ... Base 201/2,188,281,250=0.0000000B(DJ98CIF5HG60AB714E23)20 ... Base 241/858,884,438,628,757,833,498,003,410,317,638=0.000000000000000000000001D6AKMNILGKDHL265F1KJAIMFAM0(MF0B1CJD274EHGN56AKIL938)24 ... Base 301/17,344=0.001GL1(ONRA8PKEBM0QH1D562JL49FI7T3CSG)30 ... Here we are describing the integers themselves with the usual base 10 notation, and only switching to base b when we give the representations of the fractions. We follow the convention of using the letters A, B, C, ... for digits 10, 11, 12, ... in bases greater than 10. These numbers were found with a computer search, which came up empty for all the other bases that were tried. But are these really the only bases in which an integer can have this property? Question: In what bases can the reciprocal of an integer have a non-redundant pandigital reptend? Partial answer: We will prove with some simple number theory that there are no solutions for odd bases, and computer calculations have ruled out solutions for bases 8, 16, 32 and 64. But ...