## Decimal Expansion

The decimal expansion of a number is its representation in base 10. For example, the decimal expansion of is 625, of is 3.14159..., and of is 0.1111....

If has a finite decimal expansion, then

 (1)

Factoring possible common multiples gives
 (2)

where (mod 2, 5). Therefore, the numbers with finite decimal expansions are fractions of this form. The number of decimals is given by . Numbers which have a finite decimal expansion are called Regular Numbers.

Any Nonregular fraction is periodic, and has a period independent of , which is at most Digits long. If is Relatively Prime to 10, then the period of is a divisor of and has at most Digits, where is the Totient Function. When a rational number with is expanded, the period begins after terms and has length , where and are the smallest numbers satisfying

 (3)

When (mod 2, 5), , and this becomes a purely periodic decimal with
 (4)

As an example, consider .

so , . The decimal representation is . When the Denominator of a fraction has the form with , then the period begins after terms and the length of the period is the exponent to which 10 belongs (mod ), i.e., the number such that . If is Prime and is Even, then breaking the repeating Digits into two equal halves and adding gives all 9s. For example, , and . For with a Prime Denominator other than 2 or 5, all cycles have the same length (Conway and Guy 1996).

If is a Prime and 10 is a Primitive Root of , then the period of the repeating decimal is given by

 (5)

where is the Totient Function. Furthermore, the decimal expansions for , with , 2, ..., have periods of length and differ only by a cyclic permutation. Such numbers are called Long Primes by Conway and Guy (1996). An equivalent definition is that
 (6)

for and no less than this. In other words, a Necessary (but not Sufficient) condition is that the number (where is a Repunit) is Divisible by , which means that is Divisible by .

The first few numbers with maximal decimal expansions, called Full Reptend Primes, are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ... (Sloane's A001913). The decimals corresponding to these are called Cyclic Numbers. No general method is known for finding Full Reptend Primes. Artin conjectured that Artin's Constant is the fraction of Primes for with has decimal maximal period (Conway and Guy 1996). D. Lehmer has generalized this conjecture to other bases, obtaining values which are small rational multiples of .

To find Denominators with short periods, note that

The period of a fraction with Denominator equal to a Prime Factor above is therefore the Power of 10 in which the factor first appears. For example, 37 appears in the factorization of and , so its period is 3. Multiplication of any Factor by a still gives the same period as the Factor alone. A Denominator obtained by a multiplication of two Factors has a period equal to the first Power of 10 in which both Factors appear. The following table gives the Primes having small periods (Sloane's A046106, A046107, and A046108; Ogilvy and Anderson 1988).

 period primes 1 3 2 11 3 37 4 101 5 41, 271 6 7, 13 7 239, 4649 8 73, 137 9 333667 10 9091 11 21649, 513239 12 9901 13 53, 79, 265371653 14 909091 15 31, 2906161 16 17, 5882353 17 2071723, 5363222357 18 19, 52579 19 1111111111111111111 20 3541, 27961

A table of the periods of small Primes other than the special , for which the decimal expansion is not periodic, follows (Sloane's A002371).

 3 1 31 15 67 33 7 6 37 3 71 35 11 2 41 5 73 8 13 6 43 21 79 13 17 16 47 46 83 41 19 18 53 13 89 44 23 22 59 58 97 96 29 28 61 60 101 4

Shanks (1873ab) computed the periods for all Primes up to 120,000 and published those up to 29,989.

References

Conway, J. H. and Guy, R. K. Fractions Cycle into Decimals.'' In The Book of Numbers. New York: Springer-Verlag, pp. 157-163 and 166-171, 1996.

Das, R. C. On Bose Numbers.'' Amer. Math. Monthly 56, 87-89, 1949.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 159-179, 1952.

Lehmer, D. H. A Note on Primitive Roots.'' Scripta Math. 26, 117-119, 1963.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, p. 60, 1988.

Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 147-163, 1957.

Rao, K. S. A Note on the Recurring Period of the Reciprocal of an Odd Number.'' Amer. Math. Monthly 62, 484-487, 1955.

Shanks, W. On the Number of Figures in the Period of the Reciprocal of Every Prime Number Below 20,000.'' Proc. Roy. Soc. London 22, 200, 1873a.

Shanks, W. On the Number of Figures in the Period of the Reciprocal of Every Prime Number Between 20,000 and 30,000.'' Proc. Roy. Soc. London 22, 384, 1873b.

Shiller, J. K. A Theorem in the Decimal Representation of Rationals.'' Amer. Math. Monthly 66, 797-798, 1959.

Sloane, N. J. A. Sequences A001913/M4353, A002371/M4050, A046106, A046107, and A046108 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.