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Rational Exponential Expressions and a Conjecture Concerning π and e Author(s): W. S. Brown Source: The American Mathematical Monthly, Vol. 76, No. 1 (Jan., 1969), pp. 28-34 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2316782 Accessed: 09-03-2015 18:39 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions 28 RATIONAL EXPONENTIAL EXPRESSIONS [January The TheoryofNumbers,4thed.,Oxford,1960. 4. G. H. Hardyand E. M. Wright, betweenconsecutiveprimes,Quart.J. Math. Oxford, 5. A. E. Ingham,On the difference Ser.(2) 8 (1937) 255-266. Indag.Math.,12 (1950)57-58. functions, 6. L. Kuipers,Prime-representing Bull.Amer.Math. Soc., 53 (1947) 604. function, 7. W. H. Mills,A prime-representing 8. I. Niven,Functionswhichrepresentprimenumbers,Proc. Amer.Math. Soc., 2 (1951) 753-755. Proc.Amer.Math. Soc., 3 (1952) 706-712. 9. 0. Ore,On theselectionofsubsequences, C. R. Acad. Sci. Sur une formuledonnanttous les nombrespremiers, 10. W. Sierpinski, Paris,235 (1952) 1078-1079. 11. B. R. Srinivasan,Formulaeforthe nthprime,J. Indian Math. Soc., (N.S.) 25 (1961) 33-39; 26 (1962) 248. functionand an associatedformulafor the nth prime,I, II. An arithmetical 12. 35 (1962) 68-71,72-75. NorskeVid. Selsk.Forh.Trondheim, 13. C. P. Willans,On formulaeforthe nthprimenumber,Math. Gaz., 48 (1964) 413-415. 58 (1951) 616-618. thisMONTHLY, function, A prime-representing 14. E. M. Wright, J. LondonMath. Soc., 29 (1954) 63-71. functions, A class ofrepresenting 15. - RATIONAL EXPONENTIAL EXPRESSIONS AND A CONJECTURE CONCERNING 7rAND e MurrayHill,New Jersey W. S. BROWN, Bell TelephoneLaboratories, Abstract. One of the most controversialand least well defined of mathematical problemsis the problemof simplification.The recentupsurgeof interest in mechanizedmathematicshas lent new urgencyto this problem,but so farvery littlehas been accomplished.This paper attemptsto shed lighton the situation by introducingthe class of rational exponentialexpressions,definingsimplification withinthis class, and showingconstructivelyhow to achieve it. It is shown that the only simplifiedrational exponential expressionequivalent to 0 is 0 itself,provided that an easily stated conjectureis true. However the conjecture, if true, will surelybe difficultto prove, since it asserts as a special case that 'r and e are algebraicallyindependent,and no one has yet been able to prove even the much weaker conjecture that ir+e is irrational. 1. Introduction.Basically simplificationmeans the application of mathematical identitiesto transforma given expressioninto an equivalent expression satisfyingsome desired criteria. If the class of admissible expressionsis too broad, many dilemmas,ofwhich the followingis typical, arise. A basic identityforsimplificationof expressions involvingthe logarithmfunctionis (1) logZlZ2 = logz1 + log Z2 whichholds everywhereon the Riemann surfacewhose pointshave the form (2) z = rei9,r > O- <oo <0oo. This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions 1969] RATIONAL EXPONENTIAL EXPRESSIONS 29 On this surfaceone must abandon the identity (3) e2ir = 1 whichis fundamentalforsimplificationof expressionsinvolvingthe exponential function.This identitycan be saved if we definethe logarithmfunctionover a cut plane (for example, by restricting0 in (2) to the interval -7 r<0 <7r), but then (1) is lost. Surprisingly,it is not even necessaryto introducecomplexnumbersor multivalued functionsin order to get into insurmountabledifficulty.Consider the class of expressionsgeneratedfromthe integers,the real constantswrand log 2, and the real indeterminatex by application of the rational operations,the sine function,the exponential function,the absolute value function,and substitution. Richardson [1] has shown that there is no algorithmto test whetheror not a given expressionin this class is identicallyzero. Caviness [21 has shown that the same result applies to the smaller class of expressionsgenerated as above, but with log 2 and the exponentialfunctionomitted,provided that one assumes the unsolvabilityof Hilbert's tenth problem [3]. The two proofsare essentially the same except that Caviness begins with this assumption, while Richardsonbeginswitha theoremof Davis, Putnam, and Robinson [4]. This paper attempts to shed light on the simplificationproblem by introducing the class of rational exponential (REX) expressions,definingsimplification withinthis class, and showingconstructivelyhow to achieve it. The definitionissuch that distinctsimplifiedexpressionsmay be equivalent. However, it is shown that the only simplifiedREX expressionequivalent to 0 is 0 itself,provided that an easily stated conjectureis true. Unfortunately,the conjecture,if true, will surely be difficultto prove, since it asserts as a special case that 7rand e are algebraicallyindependent,and no one has yet been able to prove even the much weaker conjecture that ir+e is irrational [5]. Section 2 definesthe class of REX expressions,and definessimplificationfor expressionsin this class. The fundamentalconjectureis discussed in Section 3, the zero-equivalence theorem for simplifiedexpressionsin Section 4, and the simplificationalgorithmin Section 5. 2. Definitions.The rationalexponential(RE X) expressionsare those which are obtained by addition, subtraction,multiplication,division,and substitution startingwith the (rational) integers,the constants7rand i, some indeterminates , ZN (denoted collectively by z), and the exponential function. The .*. Zl rational exponentialfunctions are those which can be represented by REX expressions. Note that all roots of unity are REX. It is convenient to introduce the abbreviations (4) cm= exp(iir/2n) m ? 1. In particularc1 = exp(iir/2)= i. This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions 30 RATIONAL EXPONENTIAL EXPRESSIONS [January A REX expressionis equivalentto 0 if it is identically0 when viewed as an analytic functionof z. A REX expressionis equivalentto U (that is, undefined) if it involves division by a subexpressionequivalent to 0. For example, the expression 2 (z1- z1) is equivalent to 0, while the expression 3+1/2 (z1- z1) is equivalent to U. The symbol U is consideredto be a REX expression,but the constructionof more complicated expressions involving it is forbidden.Two REX expressionsare equivalentif both are equivalent to U or if theirdifference is equivalent to 0. process [6] Any REX expressioncan be transformedby a straightforward into an equivalent weakly simplifiedREX expression.A REX expressionis said to be weaklysimplifiedifit is 0 or U or ifit has the form f(exp pi, ... g(exp pi, , exp p., z, r,Coim) , exp pn, z, 7r) where (a) f and g are relativelyprimenonzeropolynomials; (If g is the polynomial ? 1, we shall permit,and indeed insist,that it be omitted,with the sign off being reversedif necessary.Althoughwe require that the polynomialf be nonzero, the numerator of (5) might neverthelessbe equivalent to 0. For example, it mightbe the expression exp(zl+z2) -exp(zi)exp(z2). In this case f is the nonzero polynomial definedbyf(a, b, c) = a-bc.) (b) the degreeoff in cor is less than the degreeof the minimalpolynomialof corn;and (c) Pi, ... , pn(On0_) are distinctweakly simplifiedREX expressionsother than 0 or U. A weakly simplifiedREX expressionis said to be simplifiedif it is 0 or U; or if it has the form(5), and , pnare simplified,and (d) pi, (e) the set {pi, * * * , p, ivr} is linearlyindependentover the rationals. The significanceof this last conditionwill become clear in Section 4. . .. 3. The fundamental conjecture. This section discusses the fundamental conjecture on which our zero-testalgorithmdepends. The simplificationalgorithmalso depends on it, but only in that no use is made of the presentlyunknownidentitieswhose existencewould be impliedby its falsity. is thattheonlyalgebraic ROUGH STATEMENT. Roughlyspeaking,theconjecture relationsinvolving7rand thez's and exponentialsofrationalexponentialexpressions are thosewhichfollowdirectly fromthefact thatrootsofunityare algebraicnumbers and fromtheidentities This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions 1969] RATIONAL EXPONENTIAL 31 EXPRESSIONS exp(0) = 1 exp(i7r) = -1 (6) exp(z1+ z2) = exp(zi) exp(Z2). PRECISE STATEMENT. Let pl, . . ., Pnbe nonzerorationalexponentialexpressions such thattheset {p1, * * I, pn, i1r} is linearlyindependentovertherationals. Then theset {exp pi, * , exp Pn,z, ir} is algebraicallyindependentovertherationals. Proof oftheConverse.Suppose the set {pi, . . ., pPn,i7 } were linearlydepen, an such that dent over the rationals. Then there would exist integersao, n = O. aoir + Eaip (7) i=l Using (6), n (8) II (exp pi)ai - a } , exp pn, and a fortiori the set {exp pi, so the set {exp pi, would be algebraicallydependent over the rationals. ,exp pn,z,7r ofthefundamental COROLLARY. Settingn= 1 and pi =1 in theprecisestatement we obtainthecorollarythatir and e are algebraicallyindependent. conjecture, 4. Zero-eqtuivalencetheorem for simplifiedREX expressions. Let p be a to 0 or U. simplifiedREX expressionotherthan 0 or U. Then p is notequtivalent Proof. Since p is simplified,we can writeit in the form(5) where (5a)-(5e) hold. The proofis by inductionon the depthof p, definedas follows.If n =0 in (5), then the depth of p is 0. Otherwisethe depth of p is max(di, * * *, dn) + 1 d, are the depths of Pi, * - *, pnrespectively. where d1, Now let d be the depth of p. If d =0, then the proofis straightforward.In the case d > 0, we assume inductivelythat the theoremis true forexpressionsof depth less than d. It follows,using (5c) and (5d), that none of the pl, * * *, Pnis equivalent to U. Thereforeneitherthe numeratornor the denominatorof (5) is equivalent to U. Viewed as a polynomialin the elementsof the set z, } {exp pi, * , exp pn71 (9) which is algebraically independent by (5e) and the fundamental conjecture, This coefficient the numeratorof (5) has, by (5a), at least one nonzerocoefficient. is a polynomialin co, (over the integers),whose degree,by (5b), is less than the degree of the minimal polynomialof o..mIt followsthat this polynomialis not equivalent to 0, and thereforethe numeratorof (5) is not equivalent to 0. A This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions 32 RATIONAL EXPONENTIAL [January EXPRESSIONS similarargumentshows that the denominatorof (5) is not equivalent to 0, and it followsimmediatelythat p is not equivalent to 0 or U. 5. Simplificationalgorithm.The purpose of this algorithmis to transforma weakly simplifiedREX expression,other than 0 or U, into a simplifiedREX expression.The given expressionhas the form(5), and conditions(5a) through (5c) are satisfied.We now definethe innermost exponentialsin a REX expression as those whose argumentsdo not involve the exponential function.The algorithmproceeds fromthe innermostexponentials outward. Step 1 (Initialize). Set k= 0, so that {ql, , qk} and {ri, , rk} denote the emptyset. Now rewrite(5) in the form , exp, P y ... r,, z c m) g(exp pi, * , exp pn, rl, * * , r, Z, 7r) ..f(exppi,.. (10) Step 2 (Begin loop withtestfor completion).At this point the expression(10), with rl,**, rk viewed as additional indeterminates,is weakly simplified. Furthermore,r1, * *, rkare abbreviations forexp ql, , exp qk respectively, and ql, * * * qk are simplified expressions of the form qi= (1) fi(ri, *.. Finally, the set {ql, * theset {ri, , rk, z, gi(r1*** , r_1, z, 7'r,CO.k) _ rr1s_z, 7rT) * k. , qk, i7r} is linearlyindependentover the rationals,and r } is algebraically independent overtherationals.Note that the pi, ..., p n may depend on the r1, rk. If n is zero in (10), then replace r1, , rkby exp ql, , exp qk respecqk playingthe role ofPi, tively.Now (10) has the form(5) withql, * pn and (5a) through(5e) are satisfied.Thereforewe are finished. Otherwise(that is, if n>0), proceed to Step 3. Step 3 (Introduceqk1 and rk+l). Let qk+l be the argumentof any innermost exponential in (10), and replace exp qt+l by the abbreviation r+l. In general, . pn.Clearly it has the form qk+l will be a subexpressionof one of the pi, Pn* , (12) qk+l = fk+,(ri) .. * gkL+l(rly , ... rk, 2, 7r CO.) , ak, z, 7r) If we replace r1, , rkby exp ql, . . , exp qk, then (12) has the form(5) with . . .*, qk playing the role of Pi, * , pn,and (Sa) through(5e) are satisfied. Thereforeqk?l is simplified. Step 4 (Test for linear dependence).Recall that the set {Iq, * * , qk, ir} iS linearlyindependentover the rationals. If qk+?can be expressedas a linear com* , qk, then we shall rewrite(10) accordingly(see Step bination of ilr and qi, Sa). Otherwise,we shall adjoin qk+l to the set (see Step 5b). Consider the linear dependence equation ql , This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions 1969] RATIONALEXPONENTIAL EXPRESSIONS (13) aoir + 33 k+1 i31 aisq =: ? whereao, , ak+1 are undeterminedrationals. Substituting(11) and (12) into (13), replacing all of the co,,by powers of ci for some sufficiently large 1, and rearrangingterms,we obtain a polynomial in rl, * * *, rk, z, wr,and wi whose degree in wxis less than the degree of the minimalpolynomialof WI and whose coefficientsare integral linear combinations of a0, * * *, aA+,. Because of the algebraic independenceof the set {ri, * rk, z, 7r},equation (13) implies that all of these coefficients vanish. Thus we obtain a set of homogeneous integral linear equations in the unknownsao, * * *, aA+4.Since the set {q1, . * q,9t iw} is linearlyindependent,ak+1 must be nonzero in any nontrivialsolution vector, and the solution space, if any, must be one dimensional,Using standard linear methods we can determinewhetheror not a solution exists, and if so, we can find it. If a solutionexists,go on to Step 5a. Otherwise,proceed to Step 5b. Step 5a (Linear dependence-replaceqk+1and rk+1). In this case we can write , qk+ (14) boiir 2co + b'q. Ic i=1 ci where bi and ci are relativelyprimeintegersfori 0, (15) q 1qi qii r; exp(q,) C' ,a epq ix q,, 1, k. Letting , * ,k we have (16) exp(ciqi ) = (r')ci, i k 1, and (17) rk+1= exp(qk+l) c* Ii (r, ) t *=1 Substitute (16) and (17) into (10), set k' k, and proceed to Step 6. Step Sb (Linear independence--adjoin qk?l). In this case the set I q... , , ir } is linearlyindependent.Now, in (10), replace r, by d! forall i 1, * , k and proceed to Step 6. Step 6 (Simplify(10) weakly).Simplify(10) weakly, treatingrI, indeterminates.The resultis an expressionof the form f(exp pt, ' * exp pnrli , Xr, ZX ) This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions k+ 1, , as 34 MATHEMATICAL NOTES [January , rt,.Note that the set wherethe p * , p' may dependon the r', conjecture. by thefundamental independent, {rl,* r** ,r, z, ir} is algebraically Step 7 (Iterate).Drop all primes,so that (10') replaces(10), and returnto Step 2. The authorwishesto thankA. J. Goldsteinformanyvaluable discussions Acknowledgment. thecourseofthiswork. throughout References of a real variable, 1. Daniel Richardson,Some unsolvableproblemsinvolvingfunctions NOTICES, Amer.Math. Soc., 13 (1966) 135. CarnegieInstituteof Technology 2. B. F. Caviness,On canonicalformsand simplification, Ph.D. Thesis,1967. Probleme,Bull. Amer.Math. Soc., 8 (1901-02)437-479. 3. David Hilbert,Mathematische 4. MartinDavis, HilaryPutnam,and JuliaRobinson,The decisionproblemforexponential equations,Ann.ofMath.,74 (1961)425. diophantine 1949,p. 84. Press,Princeton, University Numbers,Princeton 5. C. L. Siegel,Transcendental 6. Harry Pollard,The Theoryof AlgebraicNumbers,MAA, Carus MonographNo. 9, Wiley,New York,1950,p. 38. MATHEMATICAL NOTES EDITED BY DAVID DRASIN shouldbesenttoDavid Drasin,DivisionofMathematical forthisDepartment Manuscripts IN 47907. Lafayette, Sciences,PurdueUniversity, AN AREA-WIDTH INEQUALITY FOR CONVEX CURVES of California, PAULR. CHERNOFF,University Berkeley a regionofarea A. planecurve,bounding THEOREM.Let C be a closedconvex Then Letw(0)denotethewidthofC in the0-direction. (1) A <- +L w(o)w (o+ -)dO withequalityif and onlyif C is a circle. Proof.We shall employa methoddue to A. Hurwitz (see Courant [1] p. 213) to prove the isoperimetricinequality. It clearlysufficesto establish the theorem forthe case of C2curvessuch that foreach 0, O< 0 < 2ir,thereis exactlyone point, say (x(0), y(O)), at which the normal to C makes an angle 0 with the X-axis. Then, as Courant indicates,thereis a C2 periodicfunctionp(0) such that (2) x(O) = p(o) cos 0 - p'(0) sin 0 y(0) = p(o) sin 0 + p'(0) cos 0. This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:39:24 UTC All use subject to JSTOR Terms and Conditions