ON CERTAIN SUMS RELATED TO MULTIPLE DIVISIBILITY BY THE LARGEST PRIME FACTOR

Ann. Sci. Math. Québec 29 2005, no. 2, 3 45. ON CERTAIN SUMS RELATED TO MULTIPLE DIVISIBILITY BY THE LARGEST PRIME FACTOR WILLIAM D. BANKS, FLORIAN LUCA AND IGOR E. SHPARLINSKI RÉSUMÉ. Pour un nombre entier positif n, P n désigne le plus grand facteur premier de n. Nous montrons que pour deux nombres réels fixés ϑ, µ et un entier k 2, avec ϑ, 0] et µ < k, l estimation P n µ n ϑ = x +ϑ exp + o 2k µ log x log 2 x P n k n a lieu lorsque x, ou log 2 x = log log x. Avec ϑ = µ = 0 et k = 2, nous retrouvons le résultat connu à l effet que, lorsque x, le nombre d entiers positifs n x pour lesquels P n 2 n est x exp + o 2 log x log 2 x. Nous obtenons aussi des bornes supérieure et inférieure pour le nombre d entiers positifs n x pour lesquels P ϕn k ϕn, où ϕn est la fonction d Euler. ABSTRACT. For a positive integer n, let P n denote the largest prime factor of n. We show that for two fixed real numbers ϑ, µ and an integer k 2, with ϑ, 0] and µ < k, the estimate P n µ n ϑ = x +ϑ exp + o 2k µ log x log 2 x P n k n holds as x, where log 2 x = log log x. With ϑ = µ = 0 and k = 2, we recover the known result that, as x, the number of positive integers n x for which P n 2 n is x exp + o 2 log x log 2 x. We also obtain upper and lower bounds for the number of positive integers n x for which P ϕn k ϕn, where ϕn is the Euler function.. Introduction. For an integer n 2, let P n denote the largest prime factor of n, and put P =. In many situations, especially when counting integers subject to various arithmetical constraints, it can be important to understand the set of positive integers n for which P n is small relative to n, or the set of positive integers n for which P n 2 n. Such sets often contribute to the error term in a given estimate, hence it is useful to have tight bounds for the number of elements in these sets. For example, Reçu le 6 septembre 2004 et, sous forme définitive, le 3 février 2005. c Association mathématique du Québec

32 On certain sums related to multiple divisibility by the largest prime factor such sets of integers have played an important role in estimating exponential sums with the Euler function ϕn see [4, 5], in studying the average prime divisor of an integer see [3], or in evaluating various means and moments of the Smarandache function of n see [6, 9]. Positive integers with the first property that is, P n is small are called smooth, and the distribution of smooth numbers has been extensively studied. Although there is no special name for integers with the second property that is, P n 2 n, they have also been studied in the literature. For example, in [0] it has been shown that for a fixed real number r 0, the following estimate holds as x : P n 2 n P n r = x exp + o 2r + 2 log x log log x. The error term o in the above estimate can be taken to be of the form O r log log log x/ log log x. More general sums taken over integers n x with P n in a fixed arithmetic progression have been estimated in [8]. In Theorem below, we complement the result by showing that if ϑ, µ are fixed real numbers and k 2 is an integer such that ϑ, 0] and µ < k, then the following estimate holds as x : Sk, ϑ, µ, x = P n µ n ϑ P n k n = x +ϑ exp + o 2k µ log x log log x. Our method closely resembles that of [8]. In particular, taking ϑ = µ = 0 in 2, we obtain the estimate #N k x = x exp + o 2k log x log log x 3 for the cardinality of the set N k x = {n x : P n k n}. When k = 2, the bound 3 can be recovered from by taking r = 0, and it appears also in [6]. We also consider a similar question for values of the Euler function ϕn. In Theorem 2, we give upper and lower bounds for the cardinality of the sets F k x = {n x : P ϕn k ϕn}, k = 2, 3,... Finally, for an odd integer n, we define tn to be the multiplicative order of 2 modulo n, and we give upper bounds on the cardinality of the sets T k x = {n x : n odd, P tn k tn}, k = 2, 3,... 2

W. D. Banks, F. Luca and I. E. Shparlinski 33 Throughout the paper, x denotes a large positive real number. We use the Landau symbols O and o as well as the Vinogradov symbols, and with their usual meanings. The constants or convergence implied by these symbols depend only on our parameters ϑ, µ and k. For a positive integer l, we write log l x for the function defined inductively by log x = max{log x, } and log l x = log log l x for l 2, where log denotes the natural logarithm function. In the case l =, we omit the subscript to simplify the notation; however, it should be understood that all the values of all logarithms that appear are at least. 2. Preliminary results. For the results in this section, we refer the reader to Section III.5.4 in the book of Tenenbaum []. Let ρ : R 0 R denote the Dickman function. We recall that ρu = for 0 u, ρ is continuous at u = and differentiable for all u >, and ρ satisfies the difference-differential equation Thus, uρ u + ρu = 0 u >. ρu = u u u ρv dv u. For u > 0, u, let ξu denote the unique real nonzero root of the equation e ξ = + uξ. By convention, we also put ξ = 0. Lemma. The following estimate holds: ξu = logu log u + O log2 u log u u >. Lemma 2. For any fixed real number u 0 >, the following estimate holds: ρ u = ξuρu + O/u u u 0. Lemma 3. The following estimate holds: ρu = u +ou u. As usual, we say that a positive integer n is y-smooth if P n y. For all real numbers x y 2, we denote by Ψx, y the number of y-smooth integers n x: Ψx, y = #{n x : P n y}. Lemma 4. For any fixed ε > 0 and exp log 2 x 5/3+ε y x, we have when u = log x/log y. Ψx, y = xρu + O log u, log y

34 On certain sums related to multiple divisibility by the largest prime factor Lemma 5. For all x y 2, we have Ψx, y xe u/2. We also denote by Φx, y the number of integers n x for which ϕn is y-smooth: Φx, y = #{n x : P ϕn y}. By Theorem 3. of [2], the following bound holds. Lemma 6. For any fixed ε > 0 and we have when u = log x/log y. log 2 x +ε y x, Φx, y x exp + o u log 2 u Finally, let Θx, y be the number of integers n x for which tn is y-smooth: Θx, y = #{n x : P tn y}. By Theorem 5. of [2], the following bound holds. Lemma 7. For exp log x log 2 x y x we have when u = log x/log y. Θx, y x exp 2 + o u log 2 u 3. Main Results. It is easy to see that a positive integer n satisfies P n k n if and only if n = mp m k, where m = n/p n k ; therefore, the sum in 2 can be rewritten as Sk, ϑ, µ, x = P n µ n ϑ = P m µ+kϑ m ϑ. P n k n mp m k x Theorem. Let ϑ, 0] and µ < k be fixed. Then the following estimate holds as x : Sk, ϑ, µ, x = x +ϑ exp + o 2k µ log x log 2 x. Proof. Given real numbers x y 2, we put Sx, y = {n x : P n y}; thus, Ψx, y = #Sx, y. For any integer m such that mp m k x, setting p = P m we see that m = pn, where n x/p k and P n p; in other words,

W. D. Banks, F. Luca and I. E. Shparlinski 35 n Sx/p k, p. Conversely, for a prime p and an integer n Sx/p k, p, the number m = pn clearly satisfies mp m k x. We therefore have the basic identity: Sk, ϑ, µ, x = p µ+kϑ n ϑ. 4 mp m k xp m µ+kϑ m ϑ = p x n Sx/p k, p We begin by considering the contribution to 4 coming from small primes p. Let + ϑ y = exp 2 6k µ log x log 2 x. Then n ϑ p µ+kϑ p y p y p µ+kϑ n Sx/p k, p n Ψx/p k, p Ψx/p k p µ+kϑ, p t ϑ dt 5 p µ+kϑ Ψx/p k, p +ϑ p y p y n ϑ since ϑ, 0]. To estimate the last sum in 5, we define y = exp 3 log x/ log2 x and consider separately the contributions coming from primes p y and from primes p > y. For a real number z > 0, let If p y, the inequality u p log x log y u z = logx/zk log z = log x log z k. k 3 log x log 2 x k 2 log x log 2 x holds provided that x is large enough; hence, by Lemma 5, we have p y p µ+kϑ Ψx/p k, p +ϑ p y = x +ϑ exp x +ϑ p µ+kϑ p k exp + ϑ log x log 2 x + ϑ log x log 2 x p µ k x +ϑ exp + ϑ log x log 2 x p y 6 since µ k <. For primes p in the range y < p y, we remark that log u p log p log 2 x log p < log 2 x log y log 2 x2 log x = o;

36 On certain sums related to multiple divisibility by the largest prime factor hence, by Lemma 4: y <p y p µ+kϑ Ψx/p k, p +ϑ = y <p y x +ϑ x p µ+kϑ p k ρu p + O y <p y For each p y, we also have the lower bound where p µ k ρu p +ϑ. u p = log x log p k log x log y k = + ou, u = 4k µ log x /2 + ϑ 2. log 2 x log +ϑ up log p Using Lemma 3 and the fact that ρ is decreasing for large u, we deduce that ρu p +ϑ exp + o 4k µ log x log 2 x, which implies the bound p µ+kϑ Ψx/p k, p +ϑ x +ϑ exp +o 4k µ logx log 2 x. 7 y <p y Combining the estimates 6 and 7 and substituting into 5, it follows that n ϑ x +ϑ exp + o 4k µ log x log 2 x. p y p µ+kϑ n Sx/p k, p Therefore, we see that the contribution to 4 from small primes p y is negligible compared to the claimed bound on Sk, ϑ, µ, x. Next, we estimate the contribution to 4 coming from large primes p > y 2, where y 2 = exp 4k µ log x log 2 x. We have p>y 2 p µ+kϑ n Sx/p k, p n ϑ p µ+kϑ n ϑ p>y 2 /p k x/p k p µ+kϑ p>y 2 t ϑ dt p>y 2 p µ+kϑ x p k +ϑ = x +ϑ p>y 2 p µ k x +ϑ y 2 µ k+ = x +ϑ exp + o 4k µ log x log 2 x.

W. D. Banks, F. Luca and I. E. Shparlinski 37 Hence, the contribution to 4 from large primes p > y 2 is also negligible. Thus, to complete the proof of Theorem, it suffices to show that medium primes y < p y 2 make the appropriate contribution to 4; that is, we need to show that y <p y 2 p µ+kϑ n Sx/p k, p n ϑ = x +ϑ exp + o 2k µ log x log 2 x. 8 To do this, let us first fix a prime p y, y 2 ]. Using Lemma 4, we obtain the lower bound n ϑ x/p k ϑ Ψx/p k, p ρu p x+ϑ, 9 pk+kϑ where n Sx/p k, p u p = logx/pk log p = log x log p k as before. For the upper bound, we have by partial summation: n Sx/p k, p n ϑ = x/p k t ϑ d Ψt, p = t ϑ Ψt, p t=x/pk + ϑ t= ρu p p µ k x +ϑ + x/p k x/p k t ϑ Ψt, p dt t ϑ Ψt, p dt. To bound the last term, we split the integral at w = ρu p /+ϑ x/p k. For the lower range, we have w t ϑ Ψt, p dt w t ϑ dt w +ϑ = ρu p x+ϑ. 0 pk+kϑ For the upper range, writing u t,p = log t/log p, we have by Lemma 4: x/p k w x/p k log t ϑ Ψt, p dt = ρu t,p t ϑ ut,p + O dt w log p log2 x x/p k ρu w,p + O t ϑ dt ρu w,p xϑ+ log p w p k+kϑ. Since log x u p < log x log y 2 log y holds for y < p y 2, using Lemma 3 we derive that log ρu p log x log 2 x log p.

38 On certain sums related to multiple divisibility by the largest prime factor Thus, u w,p = log w log p = logx/pk + log ρu p log p ϑ log p u p + O, Using Lemma 3 again, it follows that ρu w,p = ρu p exp Ou p = ρu p exp log o x log2 x. Substituting this estimate into and combining this with 0, we obtain the upper bound n ϑ ρu p x+ϑ log p k+kϑ exp o x log2 x. n Sx/p k, p Taking into account the lower bound 9, we obtain that n Sx/p k, p n ϑ ρu p x+ϑ log p k+kϑ exp o x log2 x. 2 Now let δ = log x log 2 x /4. Put z 0 = y, and let z l = z 0 + δ l for each l. Let L denote the smallest positive integer for which z L > y 2. For each l =,..., L, let J l denote the half-open interval J l = z l, z l ]; clearly, Now let L l= p J l ρu p p µ k For every prime p J l, we have u p = log x log p k = y <p y 2 ρu p p µ k v l = u zl log x log z l + Oδ k = log x log z l k + O l L. δ log x log z l 2 L l= p J l ρu p p µ k. 3 δ = v l + O, log 2 x since log z l log y log x log 2 x. Thus, using Lemmas and 2, we obtain that ρu p = ρv l + O u p v l max ρ u t = ρv l + O δ max ρu t. t J l t J l Consequently, ρu p p µ k = ρv l p µ k + Oδ. 4 p J l p Jl Let πt denote, as usual, the number of prime numbers p t. As the asymptotic relation log t log x log 2 x = δ 2 holds uniformly for all t y, y 2 ], we have πt = t t log t + O log 2 = t + Oδ 2. 5 t log t

W. D. Banks, F. Luca and I. E. Shparlinski 39 In particular, Noting that p µ k z µ k l #J l = πz l + δ πz l = δz l log z l + Oδ. for every p J l, we obtain that p µ k #J l z µ k l p J l = δz µ k+ l + Oδ. 6 log z l Combining 6 with 4 using the resulting estimate in 3, we have therefore shown that where δ + Oδ 2 L Hl l= ρu p p µ k δ + Oδ 2 L Hl, 7 y <p y 2 Hl = ρv l z µ k+ l Now let z be chosen such that ρu z z µ k+ = log z log z l l L. l= max ρu z z µ k+ y z y 2 log z. By Lemma 3, we have for all z [y, y 2 ]: ρu z z µ k+ log x log z = exp log2 x + o 2 log z therefore, and M = ρu z z µ k+ log x log log z = + o 2 x 2k µ, = exp log z Let κ {,..., L} be chosen such that + k µ log z ; + o 2k µ log x log 2 x. Hκ = max l L Hl. We remark that for each l {,..., L} and every number z J l, we have as in our proof of 4. Thus, for z J l, ρu z = ρv l + Oδ ρv l ρu z z µ k+ log z ρu z l z µ k l. log z l

40 On certain sums related to multiple divisibility by the largest prime factor In particular, ρu zl z µ k+ L ρu y2 y µ k+ 2 = om, log z L log y 2 which shows that z J L. Therefore y < z y 2, which implies that δ + Oδ 2 Hκ ρu p p µ k δ + Oδ 2 L Hκ. y <p y 2 We also have that if z J l, then M = ρu z z µ k+ ρu zl z µ k+ l = Hl Hκ M. log z log z l We observe that 7 immediately implies that δ + Oδ 2 Hκ ρu p p µ k δ + Oδ 2 L Hκ, y <p y 2 and since L δ log x log 2 x log x log 2 x 3/4, it follows that y <p y 2 ρu p p µ k M +o, which together with 2 implies 8. This completes the proof. Theorem 2. The bounds #F k x x exp + o k log x log 3 x and #F k x x exp + o 2k + log x log 2 x hold as x. Proof. Let S denote the set of positive integers n x for which ϕn is y-smooth. Let S 2 be the set of positive integers n x such that P ϕn k ϕn and n S. For each n S 2, there exists a prime l > y such that l k ϕn. As in the proof of Theorem 5 of [7], one can show that there are at most xy k++o such positive integers n x, that is, #S 2 xy k++o. Taking y = exp k log x log 3 x, we see that Lemma 6 applies, and using the fact that #F k x #S + #S 2,

W. D. Banks, F. Luca and I. E. Shparlinski 4 we obtain the stated upper bound. For the lower bound, we remark that P ϕn k ϕn whenever n has the form n = mp ϕm k for some integer m. If m x/y k is positive and y-smooth, it follows that n = mp ϕm k mp m k x. Moreover, since P ϕm P m y, we see that for each n, there are at most y distinct representations of the form n = mp ϕm k. We now choose y = exp 2k + log x log 2 x. Taking into account that v = logx/yk log y = 2k + log x k, log 2 x using Lemmas 3 and 4 we derive that This completes the proof. #F k x Ψx/y k, yy = v +ov xy k = x exp + o 2k + log x log 2 x. Theorem 3. The bound k #T k x x exp + o log x log 2 3 x holds as x. Proof. Let S denote the set of positive integers n x for which tn is y-smooth. Let S 2 be the set of positive integers n x such that P tn k tn and n S. For each n S 2, there exists a prime l > y such that l k tn ϕn. As in the proof of Theorem 5 of [7], one can show that there are at most xy k++o such positive integers n x, that is, #S 2 xy k++o. Taking y = exp k log x log 3 x we see that Lemma 7 applies, and using the fact that #T k x #S + #S 2, we obtain the stated upper bound.

42 On certain sums related to multiple divisibility by the largest prime factor 4. Remarks. A close analysis of the arguments used in the proof of Theorem shows that if ε > 0 is fixed, then estimate of Theorem holds uniformly for all ϑ [ +ε, ε, µ < k ε and k = olog x/ log 2 x /3. Furthermore, using the Siegel-Walfisz theorem together with 5, one can easily see that the same asymptotic formula as 2 holds for Sk, ϑ, µ, a, b, x = P n µ n ϑ P n k n P n a mod b for all positive coprime integers a and b with b log x A, where A is an arbitrary constant. Moreover, since k is arbitrary in 2, it follows that if we replace the range of summation N k x by N k x\n k+ x = {n x : P n k n}, then the same asymptotic formula as 2 holds for this smaller sum. We do not give further details in this direction. One might consider a problem related to the estimation of N k x, namely that of estimating the number of positive integers n x such that p 2 j n for j =,..., k, where p >... > p k are the k largest prime factors of n. Certainly, a more careful study of #F k x perhaps in the style of our proof of Theorem should lead to stronger lower and upper bounds in Theorem 2. On the other hand, it is unlikely that one can establish unconditionally the precise rate of growth of the counting function #F k x due to the current lack of matching lower and upper bounds for Φx, y. It is reasonable to expect that the upper bound of Lemma 6 is tight; in fact, similar lower bounds for Φx, y may be derived from certain widely accepted conjectures about the distribution of smooth shifted primes see the discussion in Section 8 of [2]. Nevertheless, the strongest unconditional results on smooth shifted primes see [], for example are not nearly sufficient to establish a lower bound of the same order of magnitude as the upper bound of Lemma 6. There is no reason to doubt that #T k x = x +o ; however, at present we do not even see how to prove that #T k x x α for some fixed α > 0. Examining the integers n = 3 k one easily concludes that #T k x log x. This bound can be improved by considering powers of other small primes, but it still leaves open the question about the true order of #T k x. Finally, it would also be interesting to prove that the set Qx = {p x : p prime, P p 2 p } is infinite. This could be considered as an weak form of the conjecture that p = m 2 + holds infinitely often. Acknowledgements. Most of this work was done during a visit by the first two authors to Macquarie University; the hospitality and support of this institution are gratefully acknowledged. During the preparation of this paper, W. B. was supported in part by NSF grant DMS-0070628, F. L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I. S. was supported in part by ARC grant DP02459.

W. D. Banks, F. Luca and I. E. Shparlinski 43 Résumé substantiel en français. Pour chaque entier n 2, P n désigne le plus grand facteur premier de n. Dans plusiers situations, en particulier celles où l on compte le nombre d entiers sujet à divers contraintes arithmétiques, il est souvent important de comprendre l ensemble des entiers positifs n pour lesquels P n est petit par rapport à n, ou l ensemble des entiers positifs n pour lesquels P n 2 n. De tels ensembles souvent contribuent au terme d erreur dans une estimation. Ainsi, il est utile d avoir de très bonnes bornes sur le nombre d éléments dans ces ensembles. Par exemple, de tels ensembles d entiers jouent un rôle important dans l estimation des sommes exponentielles avec la fonction d Euler ϕn voir [4, 5], dans l étude du «diviseur premier moyen» d un entier voir [3], ou dans l évaluation de moyennes diverses et de moments de la fonction de Smarandache de n voir [6, 9]. Les entiers positifs ayant la première propriété à savoir, P n est petit sont dits lisses, et la distribution d entiers lisses a été considérablement étudiée. Malgré le fait qu aucun nom ne leur a été attribué, les entiers satisfaisant la seconde propriété à savoir, P n 2 n ont été aussi étudiés dans le corpus. Par exemple, dans [0] on a montré que pour un nombre réel fixé r 0, l estimation suivante a lieu lorsque x : P n 2 n P n r = x exp + o 2r + 2 log x log 2 x. Ici, log 2 x = log log x. Le terme d erreur o dans l estimation ci-dessus peut être pris sous la forme O r log 3 x/ log 2 x, où log 3 x = log log log x. Plus généralement, les sommes prises sur des entiers n x où P n est dans une progression arithmétique fixée ont été estimées dans [8]. Dans ce papier, nous complétons l estimation en établissant le résultat suivant : Théorème. Soient ϑ, 0] et µ < k deux nombres fixés. L estimation suivante a lieu : P n µ n ϑ = x +ϑ exp + o 2k µ log x log 2 x. 2 P n k n Notre méthode ressemble à celle dans [8]. En particulier, en prenant ϑ = µ = 0 dans 2, nous obtenons l estimation #N k x = x exp + o 2k log x log 2 x 3 pour le cardinal de l ensemble N k x = {n x : P n k n}. Lorsque k = 2, la borne 3 peut être récupérée de en prenant r = 0, et ceci apparait aussi dans [6]. Nous considérons également une question similaire pour les valeurs de la fonction d Euler ϕn, et nous donnons des bornes inférieures et supérieures pour le cardinal des ensembles F k x = {n x : P ϕn k ϕn}, k = 2, 3, 4,....

44 On certain sums related to multiple divisibility by the largest prime factor Notre résultat est le suivant : Théorème 2. Les bornes #F k x x exp + o k log x log 3 x et #F k x x exp + o 2k + log x log 2 x ont lieu lorsque x. Finalement, pour un entier impair n, nous définissons tn comme étant l ordre multiplicatif de 2 modulo n, et nous donnons des bornes supérieures sur le cardinal des ensembles T k x = {n x : n odd, P tn k tn}, k = 2, 3, 4,.... Notre résultat est le suivant : Théorème 3. La borne k #T k x x exp + o log x log 2 3 x a lieu lorsque x. REFERENCES. R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 998, 33 36. 2. W. D. Banks, J. B. Friedlander, C. Pomerance and I. E. Shparlinski, Multiplicative structure of values of the Euler function,, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, 29 47; Fields Inst. Commun., vol. 4, Amer. Math. Soc., Providence, RI, 2004. 3. W. Banks, M. Garaev, F. Luca and I. Shparlinski, Uniform distribution of the fractional part of the average prime divisor, Forum Math. 7, 885 90. 4. W. D. Banks and I. E. Shparlinski, Congruences and exponential sums with the Euler function,, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, 49 59; Fields Inst. Commun., vol. 4, Amer. Math. Soc., Providence, RI, 2004. 5. W. Banks and I. E. Shparlinski, Congruences and rational exponential sums with the Euler function, Rocky Mountain J. Math. 36 2006, 45 426. 6. K. Ford, The normal behavior of the Smarandache function, Smarandache Notions J. 0 999, 8 86. 7. J. B. Friedlander, C. Pomerance and I. E. Shparlinski, Period of the power generator and small values of Carmichael s function, Math. Comp. 70 200, 59 605. 8. A. Ivić, On sums involving reciprocals of the largest prime factor of an integer, II, Acta Arith. 7 995, 229 25. 9. A. Ivić, On a problem of Erdös involving the largest prime factor of n, Monatsh. Math. 45 2005, no., 35-46.

W. D. Banks, F. Luca and I. E. Shparlinski 45 0. A. Ivić and C. Pomerance, Estimates for certain sums involving the largest prime factor of an integer, Topics in Analytic Number Theory, vol. I-II, pp. 769 789 Budapest, 98; Colloq. Math. Soc. János Bolyai, vol. 34, North-Holland, Amsterdam, 984.. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 995. W. D. BANKS DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI COLUMBIA, MO 652 USA E-MAIL: bbanks@math.missouri.edu F. LUCA INSTITUTO DE MATEMÁTICAS UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO C.P. 58089, MORELIA MICHOACÁN, MÉXICO E-MAIL: fluca@matmor.unam.mx I. E. SHPARLINSKI DEPARTMENT OF COMPUTING, MACQUARIE UNIVERSITY SYDNEY, NSW 209, AUSTRALIA E-MAIL: igor@ics.mq.edu.au