JOURNAL OF RESEARCH of the Natio nal Bureau of Standards - B. Mathematical Scien ces Vol. 74B, No.4, Oct ober- December 1970 The Diophantine Approximation of Roots of Positive Integers* Charles F. Osgood ** (September 21, 1970) T he fo ll owing , ·es ult is es tablishe d: TH EOREM: SlIp!,ose th at k ;;. 150 and m are fixed positive integers. Th en can hold Jar at most one pai r oj rela ti ve ly prime positive intege rs p and q with q ;;' 2" ( \fin + 1 )1; . Th e n ew fea tur e of this r es u lt is th at th e lowe r bo un d o n q is g ive n ex p lic itl y and is "s m a ll. '· K eywo rd s: Di op ha ntin e approx i mation : d io phantin e e qu a ti on; effec tive CQ mputabi lit y. 1. Introduction R ece ntly Sc hinzel [4] I and Dave nport [1] have eac h obtained a res u lt of th e fo ll ow in g sort: Let a be a r ea l algeb raic numb er of degree r 2. Let s be a po siti ve r ea l numb er lar ge r than s( r ), where for Davenport r > s( r) =tr+ 0(1 ) > tr while for Sc hinzel s( r) = 3(r/2) 1/2 Th en there exists an e ffec tively co mputab le positive integer qo(a, s) such that, with at most one exce p· ti on, every pair of relatively prime integers p and q with q qo (a) sa tis fi es the inequality (Also Seppo Hyyro in [2J obtained so mething analo gous to Dave nport 's re sult for kth roots of rational numbers , where k 2 is a positive int ege r.) No ne of thes e authors, however, calc ulat ed qo (a, s) ex pli ci tl y for any cl ass of a and s. With the aid of a th eore m in a r ece nt pap er by the pr ese nt author we can obtain e xp licitly a rather "s mall" qo (a, s) for a ce rtain class of a and s. (Below one could drop the low er bo und on k very conside rably by allowing a larger s < k and a larger qo( a, s).) THEOREM I: Suppose that k 150 and III are fixed positive integers. Then (1) can hold for at most one pair of re latively prime positive integers p and q with q 2 9 (rm + 1) 6. DEFI NITIO NS: By a redu ced approximation we shall mean a fraction E where p and q are posi- q ti ve integers and (p, q)= 1. Set 8({3 ) k-2) /.L 15) . AMS s ubj ec t class ifi ca tion. Primar y IOF05. *A n inv it ed paper. Thi s paper was writt en in pari while the author was on a posldoctu.ai Reseo.;·ch Associaleship al the Na tional Bureau of Standa rds. Wash· ington. D.C. ( Thi s awa rd is give n by the National Bureau of Sta nd ards in assoc iation wi th the Nat ional Academ y of Sciences and the National Researc h Cou nc il. ) "Prese nt address: Naval Resealch Laboratory . Washi ngton, D. C. 20390. 1 Figures in b racke ts indicate the lit e ratur e refe re nces at the end of t hi s pape r. 241
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JOURNAL OF RESEARCH of the Na tional Bureau of Standards - B. Mathematical Sciences Vol. 74B, No.4, October-December 1970
The Diophantine Approximation of Roots of Positive Integers*
Charles F. Osgood**
(September 21, 1970)
T he fo llowing ,·es u lt is esta bl ished: TH EOREM: SlIp!,ose th at k ;;. 150 and m are fixed positive integers. Th en
can hold Jar at most one pair oj relatively prime positive integers p and q with q ;;' 2" ( \fin + 1 )1; . The new fea ture of thi s resu lt is th a t th e lowe r boun d o n q is g ive n ex p lic itl y and is "s m a ll. '·
K eywo rds : Di op ha ntine approx imation : d io phantine e qu a ti o n; effec t ive CQ mputabi lit y.
1. Introduction
Rece nt ly Schinzel [4] I and Dave nport [1] have each obtain ed a res u lt of the followin g so rt : Let a be a real algeb raic numbe r of degree r ~ 2. Le t s be a positive real numbe r large r than s( r), whe re for Daven port r > s( r ) = tr+ 0(1 ) > t r while for Schinzel s( r) = 3(r/2) 1/2 Th e n th ere ex is ts an e ffectively co mputable positive integer qo(a , s) s uc h th at , with a t mos t one exce p· tion, every pair of relatively prim e integers p and q with q ~ qo (a) sa tis fi es th e in equ ality
(Also Seppo Hyyro in [2J obtained something analogous to Davenport 's result for kth roo ts of rational numbers , where k ~ 2 is a positive integer.)
No ne of these authors, however , calculated qo (a, s) ex pli citl y for any class of a and s. With the aid of a th eore m in a recent paper by th e prese nt author we can obtain explicitly a ra ther "small" qo (a, s) for a ce rtain class of a and s. (Below one could drop the lower bound on k very co ns iderably by allowing a larger s < k and a larger qo( a, s).)
THEOREM I: Suppose that k ~ 150 and III are fixed positive integers. Then
(1)
can hold for at most one pair of relatively prime positive integers p and q with q ~ 29 (rm + 1) 6.
DEFI NITIO NS: By a reduced approximation we shall mean a fraction E where p and q are posiq
tive integers and (p, q)= 1. Set 8({3 ) =~ ((~ k-2) /.L 15) .
AMS s ubjec t c lassifica t ion. Primary IOF05. *A n inv it ed paper. This paper was writt en in pari while the au thor was on a posldoctu.ai Reseo.;·ch Associaleship al the Na tional Bureau of Standa rd s. Was h·
ington. D.C. (This awa rd is given by the Na t ional Bureau of Sta ndards in assoc iation wi th the Nat iona l Academ y of Sciences a nd the Na tional Research Cou nc il. )
"Prese nt address: Naval Resealch Laboratory . Washi ngton, D. C. 20390.
1 Figures in brackets indica te the lit e rature refe re nces a t the end of thi s pape r.
241
THEOREM II: If m and k ;;,: 150 are positive integers and (3 is any positive integer such that m < 28(13) then (1) is satisfied by at most (3 + 1 distinct reduced approximations.
THEOREM III: If k ;;,: 150 and m < (qot k- 12- 16)~ are positive integers then (1) is satisfied by at
most two distinct reduced fractions with denominators larger than or equal to qo ;;,: 2. Setting qo ;;,: 10 in Theorem III we obtain,
COROLLARY: If k ;;,: 150 and m are positive integers with {/ill < lQ2° then (1) is satisfied by at most
two distinct reduced fractions with denominators larger than 9.
2. Section I
PROOF OF THEOREM I. We shall use the Theorem 0/[3] (which will be stated below for the case kl=l , n =2, E= 2) and the supposed existence of two solutions of (1), p', q' and p" , q" where (p' , q') = (p", q") = 1 and q" > q' ;;,: 2 9 ( -rm + 1)6.
From [3] we have: Let sand k be positive integers , and k ;;,: 2. Let 0 < E < + 00 be a real number. Le t
Let N denote a positive integer larger than K. Set
1 > A(N) = ~Og t~~~ > 0, og 27 and
Let q denote a positive integer and (P I, P2) a nonzero vector of nonnegative integers. Let C denote any real number satisfying 0 ~ C ~ 1. Then we have, THEOREM: If q > cp(N)
Returning to the proof of Theorem I , if 1\5'ni-p'(q') - 11 < (q,) - i k then I( q')km-(p')k l k
< h( -{Y'ih + 1)H(q') 8. ChooseN to be the smaller ofm(q')k and (p')k. (Then m(q') k;;,: N > m(q')k k
-k( \5'ni + 1)"·+ I(q')8. ) Set s= I (q')""m- (p')""!.
Note that then
(2)
We shall presently show that N > K. Assuming N > K for the present and setting C = 1, we see that by the Theore m from [3] ,
l -{Y"in _ pq - I I;;,: (2q') - 1(2q) - (1+ A~N )) (3)
for all positive integers p and q with q > (27KN) 6 + 3h· - 1 • We shall eventually contradict (3) with p = rp" , q = rq" for a positive integer r ;;': 2.
First to obtain lower bounds on N. Using q' > 29 ( -{Y"in + 1)6 and k ;;,: 150 we see that
Th e las l lWO in equaliti es co me from (2) . Using the triangle in equ ality we co nclude th at
7 I' " I ' -- k P P , " 2 (q ) 8 > -;; - q" > (q q ) - I ; thus , 7
q" > t(q ')Sk- 1 'i!i. /VI I .
Now
1 +_3_= 4+ 3 log (27[(2) < 4+ 3 log (27[(2) A (N) log (N[( - l ) log [(
because N > /(2 as we saw in (4). Thus
3 3 log 27 1 1 + A(N) < 10 + log f( < 10 10 ' (5)
3. I since [( = 2"ih+'j S 2 > 2225 . Now since q' < q" and k ~ 150,
7 . (2q ') - 1 (2 q") - 10 .1 > (q") -8 k
so by th e Theorem from [3], by formula (2), and by our bound on N,
L 2 - [M M- 1] 1 d S> _ log (q") . N 1..kr,:;;_ rp"l (2 ,,) - i Sl, et r- 2 I + an u - 1 (2") ow v m ,,< rq . og rq rq
Further 8 ~ [log (q") ] [log (4M M Il q") ] - I > [log (Md ] [log (4M2) ] - 1 , since q" > 1111. We wish to show that
This would give us our desi red conlradi ction since by (3) and (5) and the inequaliti es q" > t( q' ) ~ k - I and k ~ 150
while we would have thal
I {1m - rp:: I > (2q') - 1 (2rq") - 10.1 > (2rq") - 10.2 rq
I ~ - rp:: 1< (2rq")- 10.2. rq
k 1
Since m (..,ym,+ 1 P k- 2 < (q') 22 - 4zk and k2 < 2" we see that (6) is implied by
~ He re on Ly we use the g rea tes t integer notation.
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which is implied by
But then all that we have to see is that
Htk-I)(6+3k- 1) - 1 ~ lOt
if k ~ 150. This is easily done and proves Theorem I.
3. Section II
PROOF OF THEOREM II: If ~' and p:: satish (1) and q" > q' then either p: = P;; or q" >t(q') i/H
q q q q Thus if there is a f3 + 1 - st reduced approximation (ordering them by the magnitude of their denomi-
nators) it has a denominator at Least as Large as 2 (~k_2)/3 ~ 215 ( -{Ym)6 ~ 29 ( {1m + 1) 6, so it is the finaL reduced approximation which satisfies (1).
, " PROOF OF THEOREM III: If there are two reduced fractions P, and P" with q" > q' ~ qo which
q q
satisfy (1) then
so these are all of the reduced approximations which satisfy (1).
4. References
[lJ Davenport. H. , A Note on Thue's Th eo re m. Mathematika , vol. 15,76- 87 (1968). [2J Hyyrii, Seppo, Uber Die Catalansche Problem , Ann. Acad. Scient. Fennicae, Series A, No. 355, 1-50 (1964). [3J Osgood, C. f., The Simultaneous Approximation of Certain k-th Roots , Proc. Camb. Phil. Soc. 67,75- 86 (1970). [4J Sc hin zel. Zenlralblatt fiir Math .. 137 , 258 (1967).
