Vol. 10 No 5 - Pi Mu Epsilon

Dirichlet's theorem, the arithmetic progression, where the quantity inparentheses is the base b numeral al.. .amc and c is a base b digit chosen sothat b and a]. ..amc have no common factors greater than 1,From (6) we obtain the desired result for Part (a),since Z;=,, C+l) = 2-. Note that, from Stirling's formula, (3 = 4"(n~)-'~as n + m. Consequently, we see from (7) that p,, = (nr)ln as n + m .Reference1. H. W. Gould, Combinatorial Identities, Morgantown, W. Va.,University, 1972.ALSO solved by William Chau, and Mark Evans.Editorial note: So much for a simple, little problem intended forundergraduates!864. [Fall 19951 Proposed by Charles Ashbacher, GeographicDecisions Systems, Cedar Rapids, Iowa.On page 11 of the booklet Only Problem, Not Solutions! by FlorentineSmarandache, there is the following problem.Let aI, 4, ..., am be digits. Are there primesy on a base b, whichcontain the group of digits into its writing? But n!? But n o ?Prove that for any such sequence of digits al, 4, .. , am, no matter howgenerated, there exists a prime such that the sequence is found in that prime.I. Solution by H. -J. Seiflert, Berlin, Germany.Let b > 1 be a natural number and al, . . . , am be base b digits. Bycontains infinitely many primes. Clearly, each such prime has a base brepresentation of the fom ... aI...amc. In base loy we may take c = 1.11. Comment by Bob Prielipp, University of Wisconsin-Oshkosh,Oshkosh, Wisconsin.On pages 154-155 of 131, Sierpinslci established the following result. Foran arbitrary fi~te sequence el, c2, . ., cm of digits there exists a primenumber whose fxst m digits are cI, c2, . . . , cm. George Barany [I] provedthere are inf~tely many such primes. Sierpinski [4] also proved given twoarbitrq fhte sequences of digits (of the decimal system) aI, 6,. ., a,,, andb,, b2, ..., b,,, where b,, = 1, 3, 7, or 9, there exist arbitrarily many primenumbers whose first m digits are the string a,, a2, . ., am and whose last ndigits are bl, b2, . ., bn. Borucki and Diaz [2] proved both of Sierpinski'sresults for a Dirichlet arithmetic progression.References1. American Mathematical Monthly, Advanced Problem 5738,78(1971)683.2. L. J. Borucki and J. B, Dim, "A Note on F'rimes, with Arbitrary Initialor Terminal Decimal Ciphers, in Dirichlet Arithmetic Progression,''American Mathematical Monthly, 8 l(1974) 1001- 1002.3. Waclaw Sierpinski, Elementary meory of Numbers, Hafner hblishingCompany, New York, 1964.4. Waclaw Sierpinski, A Selection of Problems in the Zheory of Numbers,The Macmillan Company, New York, 1964, page 40.Also solved by Paul S. Bruchan, Pat Costello, Thomas C. Leon;,David E. Manes, and the Proposer.