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 4. G. H. Hardyand E. M. Wright,The Theoryof Numbers, 4th ed., Oxford,1960. 5. A. E. Ingham,On the differencebetween consecutive primes, Quart. J. Math. Oxford, Ser.(2) 8 (1937) 255-266. 6. L. Kuipers,Prime-representing functions, Indag. Math., 12 (1950) 57-58. 7. W. H. Mills,A prime-representingfunction, Bull. Amer.Math. Soc., 53 (1947) 604. 8. I. Niven,Functions which represent prime numbers, Proc. Amer.Math. Soc., 2 (1951) 753-755. 9. 0. Ore,On theselection of subsequences,Proc. Amer. Math. Soc., 3 (1952) 706-712. 10. W. Sierpinski,Sur une formuledonnant tous les nombrespremiers, C. R. Acad. Sci. Paris,235 (1952) 1078-1079. 11. B. R. Srinivasan,Formulae for the nth prime,J. Indian Math. Soc., (N.S.) 25 (1961) 33-39; 26 (1962) 248. 12. An arithmeticalfunction and an associatedformula for the nth prime,I, II. NorskeVid. Selsk.Forh. Trondheim, 35 (1962) 68-71, 72-75. 13. C. P. Willans,On formulaefor the nthprime number, Math. Gaz., 48 (1964) 413-415. 14. E. M. Wright,A prime-representingfunction, this MONTHLY, 58 (1951) 616-618. 15. - A class of representingfunctions, J. LondonMath. Soc., 29 (1954) 63-71. RATIONAL EXPONENTIAL EXPRESSIONS AND A CONJECTURE CONCERNING 7rAND e W. S. BROWN, Bell TelephoneLaboratories, Murray Hill, New Jersey Abstract. One of the most controversialand least well defined of mathe- matical problemsis the problemof simplification.The recentupsurge of interest in mechanizedmathematics has lent new urgencyto this problem,but so farvery little has been accomplished.This paper attempts to shed light on the situation by introducingthe class of rational exponentialexpressions, defining simplifica- tion withinthis class, and showingconstructively how to achieve it. It is shown that the only simplifiedrational exponential expressionequivalent to 0 is 0 it- self,provided that an easily stated conjecture is true. However the conjecture, if true, will surely be 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 mathe- matical identitiesto transforma given expressioninto an equivalent expression satisfyingsome desired criteria. If the class of admissible expressionsis too broad, many dilemmas,of which the followingis typical, arise. A basic identityfor simplificationof expressions involvingthe logarithmfunction is (1) logZlZ2 = log z1 + log Z2 whichholds everywhereon the Riemann surfacewhose points have 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 which is fundamentalfor simplification of expressionsinvolving the exponential function.This identitycan be saved if we definethe logarithmfunction over 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 introducecomplex numbers or multi- valued functionsin order to get into insurmountabledifficulty. Consider the class of expressionsgenerated from the integers,the real constantswr and log 2, and the real indeterminatex by application of the rational operations,the sine function,the exponential function,the absolute value function,and substitu- tion. 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 expressions generated as above, but with log 2 and the exponential functionomitted, provided that one assumes the unsolvabilityof Hilbert's tenth problem [3]. The two proofs are essentially the same except that Caviness begins with this assumption, while Richardsonbegins with a theoremof Davis, Putnam, and Robinson [4]. This paper attempts to shed light on the simplificationproblem by intro- ducing the class of rational exponential (REX) expressions,defining simplifica- tion withinthis class, and showingconstructively how to achieve it. The definitionissuch that distinctsimplified expressions may be equivalent. However, it is shown that the only simplifiedREX expressionequivalent to 0 is 0 itself,provided that an easily stated conjecture is true. Unfortunately,the conjecture, if true, will surely be difficultto prove, since it asserts as a special case that 7rand e are algebraically independent,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 definessimplification for expressionsin this class. The fundamentalconjecture is discussed in Section 3, the zero-equivalence theorem for simplifiedexpressions in Section 4, and the simplificationalgorithm in Section 5. 2. Definitions.The rational exponential(RE X) expressionsare those which are obtained by addition, subtraction,multiplication, division, and substitution startingwith the (rational) integers,the constants7r and i, some indeterminates Zl .*. , ZN (denoted collectively by z), and the exponential function. The 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 ex- pression 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. Any REX expressioncan be transformedby a straightforwardprocess [6] into an equivalent weakly simplifiedREX expression.A REX expressionis said to be weaklysimplified if it is 0 or U or ifit has the form f(exp pi, ... , exp p., z, r,Coim) g(exp pi, , exp pn, z, 7r) where (a) f and g are relativelyprime nonzero polynomials; (If g is the polynomial ? 1, we shall permit,and indeed insist,that it be omitted,with the sign off being reversedif necessary. Although we re- quire 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 degree off in cor is less than the degree of the minimalpolynomial of 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 (d) pi, . .. , pnare simplified,and (e) the set {pi, * * * , p, ivr} is linearlyindependent over 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-testalgorithm depends. The simplificationalgo- rithmalso depends on it, but only in that no use is made of the presentlyun- knownidentities whose existencewould be implied by its falsity. ROUGH STATEMENT. Roughlyspeaking, the conjecture is thatthe only algebraic relationsinvolving 7r and thez's and exponentialsof rational exponential expressions are thosewhich follow directly from the fact thatroots of unity are algebraicnumbers and fromthe identities 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 31 exp(0) = 1 (6) exp(i7r) = -1 exp(z1+ z2) = exp(zi) exp(Z2). PRECISE STATEMENT. Let pl, . ., Pnbe nonzerorational exponential expres- sions such thatthe set {p1, * * I, pn, i1r} is linearlyindependent over the rationals. Then theset {exp pi, * , exp Pn,z, ir} is algebraicallyindependent over the ra- tionals. Proof of theConverse. Suppose the set {pi, . ., pPn,i7 } were linearlydepen- dent over the rationals. Then there would exist integersao, , an such that n (7) aoir + Eaip = O. i=l Using (6), n a (8) II (exp pi)ai - so the set {exp pi, , exp pn}, and a fortiorithe set {exp pi, , exp pn,z,7r would be algebraically dependent over the rationals. COROLLARY.Setting n= 1 and pi =1 in theprecise statement of the fundamental conjecture,we obtainthe corollary that ir and e are algebraicallyindependent. 4. Zero-eqtuivalencetheorem for simplifiedREX expressions. Let p be a simplifiedREX expressionother than 0 or U. Then p is not equtivalentto 0 or U.