166 History of the Theory of Numbers. [Chap, vi 
W. Stammer^^ noted that n/p = 0.di . . . a^ implies 
-(10'-l)=ai...a,. 
V 
J. B. Sturm^^ used this result to explain the conversion of decimal into 
ordinary fractions without the use of series. 
M. Collins'^^ stated that, if we multiply any decimal fraction having m 
digits in its period by one with n digits, we obtain a product with Own 
digits in its period if vi is prime to n, but with 71(10™ — 1) digits if n is 
divisible by m. 
J. E. Oliver^^ proved the last theorem. If x'/x gives a periodic fraction 
to the base a with a period of ^ figures, then a^ = 1 (mod x) and conversely. 
The product of the periodic fractions for x'/x, . . . , z'/z with period lengths 
^, . . . , f has the period length 
•M(^,...,f), 
M{x,...,z) 
where M{x, . . ., z) is the 1. c. m. of x, . . . , z. He examined the cases in 
which the first factor in the formula is expressible in terms of ^, . . . , f . 
Fr. Heime'*^ and M. Pokorny^^ gave expositions without novelty. 
Suffield*^ gave the more important rules for periodic decimals and indi- 
cated the close connection with the method of synthetic division. 
W. H. H. Hudson*^ called d a proper prime if the period for n/d has d — 1 
digits. If the period for r/p has n = ip — l)/\ digits, there are X periods 
for p. The sum of the digits in the period for a proper prime p is 9{p — 1)/2. 
If 1/p has a period of 2n digits, the sum of corresponding digits in the two 
half periods is 9, and this holds also if p is composite but has no factor 
dividing 10" — 1 [Midy"]. If lOp+l is a proper prime, each digit 0, 1, . . . , 9 
occurs p times in its period. If a, h are distinct primes with periods of 
a, /3 digits, the number of digits in the period for ab is the 1. c. m. of a, /8 
[Bernoulli^]. Let p have a period of n digits and l/p = A-/(10" — 1). Let m 
be the least integer for which 
\ljp^-'^\2)p^-''^- ^\x-l) p 
is an integer; then 1/p^ has a period of mn digits. 
"Archiv Math. Phys., 27, 1856, 124. 
«/6id., 33, 1859, 94-95. 
«Math. Monthly (ed., Runkle), Cambridge, Mass., 1, 1859, 295. 
**Ibid., 345-9. 
♦'Ueber relative Prim- und correspondirende Zahlen, primitive und sekundare Wurzeln und 
periodische Decimalbriiche, Progr., BerUn, 1860, 18 pp. 
"Ueber einige Eigenschaften periodischer Dezimalbriiche, Prag, 1864. 
♦^Synthetic division in arithmetic, with some introductory remarks on the period of circulating 
decimals, 1863, pp. iv-|-19. 
*80xford, Cambridge and Dublin Messenger of Math., 2, 1864, 1-6. Glaisher" atrributed this 
useful anonymous paper to Hudson. 
