probability of directional errors with disordinal qualitative interaction in a factorial design with two or more factors there is nonzero interaction when the differences among the levels of one factor vary with levels of other factors the interaction is disordinal or qualitative with respect to a specific factor saya if the difference between at least two levels ofa is positive for some and negative for some levels of the other factors using standard methods of analysis there is a potentially large probability of drawing incorrect conclusions about the signs of differences in the presence of disordinal interaction the maximum probability of such incorrect conclusions or directional errors is derived for two factor designs in which the factor of interest has two levels and the number of levels of the other factor varies

a primitive trinomial of degree 6972593 the only primitive trinomials of degree 6972593 over gf 2 are x 6972593 x 3037958 1 and its reciprocal

three new factors of fermat numbers we report the discovery of a new factor for each of the fermat numbers f 13 f 15 f 16 these new factors have 27 33 and 27 decimal digits respectively each factor was found by the elliptic curve method after division by the new factors and previously known factors the remaining cofactors are seen to be composite numbers with 2391 9808 and 19694 decimal digits respectively

a likelihood ratio test regarding two nested but oblique order restricted hypotheses abstract cadoret woolson and winokur 1977 considered two theories regarding the genetic makeup of patients suffering from unipolar affective disorder these two theories imply nested but oblique order restrictions on the parameters of a statistical model a likelihood ratio test for these two restrictions is studied and found to be dominated by a test that neglects some of the information in the model

an optimal control of water level considering time delay system this paper presents a numerical application of the optimal control considering time delay system to the flood control with drainage basin the optimal control theory is used to obtain the control values which satisfies the state equation and minimizes performance function feature of this study is application of time delay system to control problems two kinds of control values which has time difference can be introduced those are quantity of water intake and release in drainage basin these control values are correlated with each other which can be modeled by the time delay system the sakawa shindo method is employed for the minimization algorithm to calculate the water flow phenomenon the shallow water equation is employed the clank nicolson method is applied to temporal discretization and the stabilized bubble function method is applied to spatial discretization as the numerical model the flood flow in the tsurumi river which locates in yokohama city is represented

upper bounds for the prime divisors of wendt s determinant let c 2 be an even integer 3 c 1 the resultant wc of the polynomials tc 1 and 1 t c 1 is known as wendt s determinant of order c we prove that among the prime divisors q of wc only those which divide 2c 1 or lc 2 can be larger than c 4 where 2 2487338 and ln is the nth lucas number except when c 20 and q 61 using this estimate we derive criteria for the nonsolvability of fermat s congruence

improved techniques for lower bounds for odd perfect numbers if n is an odd perfect number and qk n q prime k even then it is almost immediate that n q2k we prove here that subject to certain conditions verifiable in polynomial time in fact n q5k 2 using this and related results we are able to extend the computations in an earlier paper 1 to show that n 10300 comments only the abstract is given here the full paper appeared as 2 the main part of the proof that there is no odd perfect number n less than 10300 is a very large tree each of whose 12655 leaves gives either a contradiction or a sufficiently large lower bound on n the proof tree is available by anonymous ftp in a separate file rpb116p dvi z the integer factorizations used in the proof are also available by anonymous ftp see 3 references 1 r p brent and g l cohen a new lower bound for odd perfect numbers mathematics of computation 53 1989 431 437 supplement ibid s7 s24 mr 89m 11008 also appeared as report tr cs 88 05 computer sciences laboratory anu february 1988 50 pp rpb100 2 r p brent g l cohen and h j j te riele improved techniques for lower bounds for odd perfect numbers mathematics of computation 57 1991 857 868 mr 92c 11004 also appeared as report nm r8921 centrum voor wiskunde en informatica amsterdam october 1989 13 pp a longer version was submitted to the mathematics of computation umt file and appeared as report cma r50 89 cma anu october 1989 198 pp rpb116 3 r p brent factor an integer factorization program for the ibm pc report tr cs 89 23 computer sciences laboratory anu october 1989 7 pp rpb117 brent computer sciences laboratory australian national university canberra e mail address rpb cslab anu edu au cohen school of mathematical sciences university of technology sydney po box 123 broadway nsw 2007 australia te riele centrum voor wiskunde en informatica kruislaan 413 1098 sj amsterdam the nether lands e mail address herman cwi nl 1991 mathematics subject classification primary 11a25 secondary 11 04 11y05 11y70

two new tests associated with analysis of variance summary two new tests are described in connexion with analysis of variance the first is for the presence of qualitative interaction between a treatment and an intrinsic factor both with qualitatively defined levels such an interaction is said to be present if there are two levels of the treatment factor and two levels of the intrinsic factor such that the treatment difference has opposite signs at the two levels of the intrinsic factor the second test is for the presence of a treatment effect which can be tested via any of a hierarchical sequence u1 u1 u2 u1 ur of sums of squares an allowance for selection is calculated

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

figures of merit for digital multistep pseudorandom numbers the statistical independence properties of s successive digital multistep pseudorandom numbers are governed by the figure of merit p s i which depends on s and the characteristic polynomial f of the recursion used in the generation procedure we extend previous work for s 2 and describe how to obtain large figures of merit for s 2 thus arriving at digital multistep pseudorandom numbers with attractive statistical independence properties tables of figures of merit for s 3 4 5 and degrees 32 are included

hypothesis testing for marginal probabilities in a 2 2 2 contingency table with conditional independence abstract consider a 2 2 2 contingency model with factors a b and c assume factors a and b are conditionally independent given factor c we are interested in testing the null hypothesis that the marginal probabilities of the second level of b are the same across the levels of a small sample and large sample tests are studied

on the existence of optimum cyclic burst correcting codes it is shown that for each integer b geq 1 infinitely many optimum cyclic b burst correcting codes exist i e codes whose length n redundancy r and burst correcting capability b satisfy n 2 r b 1 1 some optimum codes for b 3 4 and 5 are also studied in detail

nonequivariant simultaneous confidence intervals less likely to contain zero abstract we present a procedure for finding simultaneous confidence intervals for the expectations j n j 1 of a set of independent random variables identically distributed up to their location parameters that yields intervals less likely to contain zero than the standard simultaneous confidence intervals for many 0 the procedure is defined implicitly by inverting a nonequivariant hypothesis test with a hyperrectangular acceptance region whose orientation depends on the unsigned ranks of the components of then projecting the convex hull of the resulting confidence region onto the coordinate axes the projection to obtain simultaneous confidence intervals implicitly involves solving n sets of linear inequalities in n variables but the optima are attained among a set of at most n 2 such sets and can be found by a simple algorithm the procedure also works when the statistics are exchangeable but not independent and can be extended to cases where the inference is based on statistics for

input disturbance predictive control of shallow water flows abstract the purpose of this paper is to describe the predictive control of flood flows over the surface of reservoirs in a mountainous area with a complicated terrain shape it is expected that damage to villages around the reservoir can be reduced if the water elevation of water discharge is controlled using an artificially constructed aqueduct the control method uses the first order adjoint equation and the numerical technique used is the finite element method in order to control water elevation prediction of input discharge is necessary and optimal discharges can accordingly be determined predictive control of water flows is carried out in this study the adjoint equation method is applied to obtain the optimal control a performance function is used as the index of the optimal control which is defined by the sum of the square of the computed water elevation at the target points as a minimization technique the weighted gradient method is applied the gradient of the extended performance function with respect to the control of water discharge is obtained by solving the first order adjoint equation the predictive control method employed in this paper is a technique to make prediction of the input disturbance and compute the optimal control value at a certain time duration the shallow water equation based on the water discharge and elevation is used as the governing equation as the spatial discretization the finite element method based on the linear function interpolation is employed as the temporal discretization the two step explicit scheme is applied after verification based on the predictive control of the water elevation in simple rectangular channel and sinusoidal water elevation model the present method is applied to flood control of the ikari reservoir located in the tochigi prefecture in japan

order restricted inferences in linear regression abstract regression analysis constitutes a large portion of the statistical repertoire in applications in cases where such analysis is used for exploratory purposes with no previous knowledge of the structure one would not wish to impose any constraints on the problem but in many applications we are interested in curve fitting with a simple parametric model to describe the structure of a system with some prior knowledge about the structure an important example of this occurs when the experimenter has a strong belief that the regression function changes monotonically with some or all of the predictor variables in a region of interest the analyses needed for statistical inferences under such constraints are nonstandard considering the present body of knowledge developed for unconstrained regression it will be an enormous task to derive the analogs of even a small fraction of this for the restricted case in this article we initiate the study with simple linear regression on a single variable the est

new primitive t nomials t e3 5 over gf 2 whose degree is a mersenne exponent all primitive trinomials over gf 2 with degree 859433 which is the 33rd mersenne exponent are presented they are x 859433 x 288477 1 and its reciprocal also two examples of primitive pentanomials over gf 2 with degree 86243 which is the 28th mersenne exponent are presented the sieve used is briefly described

uniformly more powerful tests for hypotheses concerning linear inequalities and normal means abstract this article considers some hypothesis testing problems regarding normal means in these problems the hypotheses are defined by linear inequalities on the means we show that in certain problems the likelihood ratio test lrt is not very powerful we describe a test that has the same size as the lrt and is uniformly more powerful the test is easily implemented since its critical values are standard normal percentiles the increase in power with the new test can be substantial for example the new test s power is 1 2 times bigger 10 times bigger for 05 than the lrt s power for some parameter points in a simple example specifically let x x 1 xp p 2 be a multivariate normal random vector with unknown mean 1 p and known nonsingular covariance matrix we consider testing the null hypothesis h 0 b i 0 for some i 1 k versus the alternative hypothesis h 1 b i 0 for all i 1 k here b 1 b k k 2 are specified p dimensional ve

tests against qualitative interaction exact critical values and robust tests summary consider a study to evaluate treatment a with a placebo in two or more groups of patients if treatment a is beneficial to one group of patients and harmful to another then we say that there is qualitative interaction or crossover interaction between patient groups and the treatments gail and simon 1985 biometrics41 361 372 developed a large sample procedure for this testing problem their test has received favorable coverage in the literature in this article we obtain corresponding exact finite sample results for normal error distribution and provide a table of critical values the test statistic is similar to the familiar f ratio and its p value is equal to a weighted sum of tail areas of f distributions the computations to implement this are simple a simulation study shows that the exact critical values provided here for normal error distribution are preferable to the asymptotic critical values for a wide range of error distributions we also develop tests that are power robust against long tailed error distributions our robust test uses m estimators instead of the least squares estimators we show that the efficiency robustness of the m estimator translates to power robustness of the corresponding test therefore our robust tests are better if outliers are expected a simulation study illustrates the substantial power advantages of our robust tests

gauss periods orders and cryptographical applications experimental results on the multiplicative orders of gauss periods in finite fields are presented these results indicate that gauss periods have high order and are often primitive self dual normal elements in finite fields it is shown that gauss periods can be exponentiated in quadratic time an application is an efficient pseudorandom bit generator

on trinomials xn x 2 1 and x8 3 xk 1 irreducible over gf 2 we list the 19 values of n for which the trinomial t n 2 x n x 2 1 is irreducible over gf 2 with n 60 000 we then show how to produce from these polynomials and their periods and indices all irreducible trinomials t n k x n x k 1 with n 3 mod 8 these are members of infinite classes of irreducible trinomials t nr 2r where r has all of its prime factors divisors of the period of t n 2 and none of its prime factors divisors of the index of t n 2

uniformly more powerful one sided tests for hypotheses about linear inequalities let x have a multivariate p dimensional normal distribution p 2 with unknown mean and known nonsingular covariance consider testing h 0 b i 0 for some i 1 k versus h 1 b i 0 for all i 1 k where b 1 b k k 2 are known vectors that define the hypotheses for any 0 1 2 we construct a size test that is uniformly more powerful than the size a likelihood ratio test lrt the proposed test is an intersection union test other authors have presented uniformly more powerful tests under restrictions on the covariance matrix and on the hypothesis being tested our new test is uniformly more powerful than the lrt for all known nonsingular covariance matrices and all hypotheses so our results show that in a very general class of problems the lrt can be uniformly dominated

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

table of primitive binary polynomials ii for those n 5000 for which the factorization of 2n 1 is known the first primitive trinomial if such exists and a randomly generated primitive 5and 7 nomial of degree n in gf 2 are given if the respective entry is absent from the previously published table after our paper with a list of primitive binary polynomials 3 had been accepted for publication s s wagstaff jr made available to us reference 2 with some new factorizations of 2n 1 updates to 1 obtained up to april 5 1993 this enabled us to make an update to our list in which the smallest degree of missing primitive binary polynomial is now 51 1 here like in our previous table the degrees of nonzero coefficients except 0 and n of primitive k nomials if there are any of degree n are given for some missing 2 n 5000 table 1 primitive binary polynomials

testing hypotheses concerning unions of linear subspaces abstract the likelihood ratio test lrt for hypotheses concerning unions of linear subspaces is derived for the normal theory linear model a more powerful test an intersection union test is proposed for the case in which the subspaces are not all of the same dimension a theorem is proved that may be used to identify hypotheses that concern unions of linear subspaces some hypotheses about the spacings between normal means are shown to concern unions of linear subspaces and therefore can be tested using the lrt finally the computation of the lrt statistic is discussed

a comparison of the power of two tests for qualitative interactions qualitative or crossover interactions arise when a new treatment compared with a control treatment is beneficial in some subsets of patients and harmful in other subsets we present a new range test for crossover interactions and compare it with the likelihood ratio test developed by gail and simon the range test has greater power when the new treatment is harmful in only a few subsets whereas the likelihood ratio test has greater power when the new treatment is harmful in several subsets we provide power tables for both tests to facilitate sample size calculations for designing experiments to detect qualitative interactions and for interpreting the results of clinical trials

prime clusters and cunningham chains we discuss the methods and results of a search for certain types of prime clusters in particular we report specific examples of prime 16 tuplets and cunningham chains of length 14

a table of primitive binary polynomials for those n 5000 for which the factorization of 2n 1 is known the first primitive trinomial if such exists and a randomly generated primitive 5 and 7 nomial of degree n in gf 2 are given a primitive polynomial of degree n over gf 2 is useful for generating a pseudo random sequence of n tuples of zeros and ones see 8 if the polynomial has a small number k of terms then the sequence is easily computed but for cryptological applications correlation attack see 5 it is often necessary to have the primitive polynomials with k larger than one can find in the existing tables for example zierler and brillhart 10 11 have calculated all irreducible trinomials of degree n 1000 with the period for some for which the factorization of 2n 1 is known stahnke 7 has listed one example of a trinomial or pentanomial of degree n 168 zierler 12 has listed all primitive trinomials whose degree is a mersenne exponent 11213 m23 here mj denotes the jth mersenne exponent rodemich and rumsey 6 have listed all primitive trinomials of degree mj 12 j 17 kurita and matsumoto 2 have listed all primitive trinomials of degree mj 24 j 28 and one example of primitive pentanomials of degree mj 8 j 27 1980 mathematics subject classification 1985 revision primary 11t06 11t71

uniformly more powerful tests in a one sided multivariate problem abstract we consider a testing problem in which a normal mean vector is at the origin under the null hypothesis and in a cone under the alternative and the covariance matrix is unknown perlman studied the likelihood ratio lr test and found that it is biased and hence does not compare favorably with hotelling s t2 we point out that certain tests of perlman are uniformly more powerful as well as simpler than the lr test these are the corresponding lr tests when the alternative is replaced with a half space that contains the cone we prove that these tests are unbiased by developing some basic results on stochastic ordering critical values for these tests are given simulation results suggest that the new tests are also uniformly more powerful than hotelling s t2 for the special case when the cone is the positive orthant we discuss an issue about invariant test finally we discuss the assumption of unknown covariance matrix and conclude that the new tests are more useful for small samples than for

efficient detection of random coefficients in autoregressive models the problem of detecting randomness in the coefficients of an ar p model that is the problem of testing ordinary ar p dependence against the alternative of a random coefficient autoregressive rcar p model is considered a nonstandard lan property is established for rcar p models in the vicinity of ar p ones two main problems arise in this context the first problem is related to the statistical model itself gaussian assumptions are highly unrealistic in a nonlinear context and innovation densities should be treated as nuisance parameters the resulting semiparametric model however appears to be severely nonadaptive in contrast with the linear arma case pseudo gaussian likelihood methods here are invalid under non gaussian densities even the innovation variance cannot be estimated without a strict loss of efficiency this problem is solved using a general result by hallin and werker which provides semiparametrically efficient central sequences without going through explicit tangent space calculations the second problem is related to the fact that the testing problem under study is intrinsically one sided while the case of multiparameter one sided alternatives is not covered by classical asymptotic theory under lan a concept of locally asymptotically most stringent somewhere efficient test is proposed in order to cope with this one sided nature of the problem

accurate critical constants for the one sided approximate likelihood ratio test of a normal mean vector when the covariance matrix is estimated summary tang gnecco and geller 1989 biometrika76 577 583 proposed an approximate likelihood ratio alr test of the null hypothesis that a normal mean vector equals a null vector against the alternative that all of its components are nonnegative with at least one strictly positive this test is useful for comparing a treatment group with a control group on multiple endpoints and the data from the two groups are assumed to follow multivariate normal distributions with different mean vectors and a common covariance matrix the homoscedastic case tang et al derived the test statistic and its null distribution assuming a known covariance matrix in practice when the covariance matrix is estimated the critical constants tabulated by tang et al result in a highly liberal test to deal with this problem we derive an accurate small sample approximation to the null distribution of the alr test statistic by using the moment matching method the proposed approximation is then extended to the heteroscedastic case the accuracy of both the approximations is verified by simulations a real data example is given to illustrate the use of the approximations

new cullen primes numbers of the forms c n n 2 n 1 and w n n 2 n 1 are both called cullen numbers new primes c n are presented for n 4713 5795 6611 18496 for w n several new primes are listed the largest one having n 18885 furthermore all efforts made to factorize numbers c n and w n are described and the result the complete factorization for all n 300 is given in a supplement

factors of generalized fermat numbers generalized fermat numbers have the form f b m b 2m 1 their odd prime factors are of the form k 2 n 1 k odd n m it is shown that each prime is a factor of some f b m for approximately 1 k bases b independent of n divisors of generalized fermat numbers of base 6 base 10 and base 12 are tabulated three new factors of standard fermat numbers are included

finding suitable curves for the elliptic curve method of factorization using the parametrizations of kubert we show how to produce in nite families of elliptic curves which have prescribed nontrivial torsion over q and rank at least one these curves can be used to speed up the ecm factorization algorithm of lenstra we also brie y discuss curves with complex multiplication in this context

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

a fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377 the standard algorithm for testing reducibility of a trinomial of prime degree r over gf 2 requires 2r o 1 bits of memory we describe a new algorithm which requires only 3r 2 o 1 bits of memory and significantly fewer memory references and bit operations than the standard algorithm if 2r 1 is a mersenne prime then an irreducible trinomial of degree r is necessarily primitive we give primitive trinomials for the mersenne exponents r 756839 859433 and 3021377 the results for r 859433 extend and correct some computations of kumada et al the two results for r 3021377 are primitive trinomials of the highest known degree

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

efficient computer search of large order multiple recursive pseudo random number generators utilizing some results in number theory we propose an efficient method to speed up the computer search of large order maximum period multiple recursive generators mrgs we conduct the computer search and identify many efficient and portable mrgs of order up to 25 013 which have the equi distribution property in up to 25 013 dimensions and the period lengths up to 10 2 3 3 3 6 1 approximately in addition a theoretical test is adopted to further evaluate and compare these generators an extensive empirical study shows that these generators behave well when tested with the stringent crush battery of the test package testu01

primality testing using elliptic curves we present a primality proving algorithm a probablistic primality test that produces short certificates of primality on prime inputs we prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes as a corollary we obtain an algorithm for generating large certified primes with distribution statistically close to uniform under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size the test is shown to run in expected polynomial time for all primes yielding a las vegas primality test our test is based on a new methodology for applying group theory to the problem of prime certification and the application of this methodology using groups generated by elliptic curves over finite fields we note that our methodology and methods have been subsequently used and improved upon most notably in the primality proving algorithm of adleman and huang using hyperelliptic curves and in practical primality provers using elliptic curves

conditional likelihood ratio test for a nonnegative normal mean vector abstract the conditional likelihood ratio test is derived for significance of a multivariate mean having nonnegative components this test is shown to be uniformly more powerful than the unconditional likelihood ratio test derived by perlman the computation involved in the new test is a straightforward programming task simulation results suggest that this test is also uniformly more powerful than a half space test proposed by tang and hotelling s t 2 test the consistency invariance and unbiasedness of the new test are established and the test is illustrated with data from a randomized clinical trial

corrigenda to new primitive nomials 3 5 over 2 whose degree is a mersenne exponent and some new primitive pentanomials

ten new primitive binary trinomials we exhibit ten new primitive trinomials over gf 2 of record degrees 24 036 583 25 964 951 30 402 457 and 32 582 657 this completes the search for the currently known mersenne prime exponents

on computing factors of cyclotomic polynomials for odd square free n 1 the cyclotomic polynomial n x satisfies the identity of gauss 4 n x an 1 n 1 2nb2 n a similar identity of aurifeuille le lasseur and lucas is n 1 x c n nxd n or in the case that n is even and square free n 2 x c n nxd n here an x dn x are polynomials with integer coefficients we show how these coefficients can be computed by simple algorithms which require o n arithmetic operations and work over the integers we also give explicit formulae and generating functions for an x dn x and illustrate the application to integer factorization with some numerical examples 1991 mathematics subject classification primary 11 04 05a15 secondary 11t06 11t22 11t24 11y16 12 04 12e10 12y05

on the problem of interactions in the analysis of variance abstract for the two way analysis of variance without the assumption of zero interactions consider the following recommendation test the zero interactions hypothesis and if the hypothesis is not rejected proceed as if the interactions were zero this recommendation is unwarranted and may give incorrect decisions with a large probability as has been pointed out in several articles beginning in the year 1935 despite such isolated criticism it is a recommendation suggested in most applied books in the field and used in countless applications an apology for the recommendation is that it can be easily improved by considering the power of the test and thus obtaining information on the interactions neglected we consider this improved recommendation for the goals of i an interval estimate of one of the cell expectations ii a simultaneous interval estimate of the cell expectations and iii an estimate of the cell with the largest expectation we show that the improved recommendation leads to method

repunit r49081 is a probable prime the repunit r49081 1049081 1 9 is a probable prime in order to prove primality r49080 must be approximately 33 3 factored the status of this factorization is included

qualitative interactions in multifactor studies in clinical trials qualitative interaction or crossover interaction is said to occur when one treatment is superior for some sets of patients and the alternative treatment is superior for other subsets here we propose a definition of no qualitative interaction with respect to a single continuous covariate which implies that one treatment is superior to the other treatment over a prespecified range of the covariate further in studies involving patients cross classified by two or more prognostic factors we define a marginal qualitative interaction with respect to a treatment factor and a prognostic factor by averaging in some sense over the remaining prognostic factors in some situations the methodology of gail and simon 1985 biometrics 41 361 372 is shown to be appropriate their procedure assumes independent estimates of treatment effect for each subset of patients and this may not be appropriate for the generalizations we propose therefore we generalize the procedure of gail and simon to the case of two correlated estimates of treatment effect providing a table of critical values results for one sided tests for the case of j correlated estimates are also obtained we also present an example illustrating our procedure

computing special powers in finite fields we study exponentiation in nonprime finite fields with very special exponents such as they occur for example in inversion primitivity tests and polynomial factorization our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time

primitive normal bases for finite fields it is proved that any finite extension of a finite field has a normal basis consisting of primitive roots introduction let q be a prime power q 1 we denote by f9 a finite field of q elements it is well known that for every positive integer m there exists a normal basis of f m over f9 i e a basis of the form q q 2 qp aaq aq2 a9 1 with a e f m it is also well known that the multiplicative group f of fqm is cyclic i e that for some a e pq we have f an n e z such an element a is called a primitive root of fqm following davenport 4 we call qq2 rn i a normal basis a a a9 a a a9 of fq over fq a primitive normal basis if a is a primitive root of fq carlitz 2 3 proved in 1952 that for all sufficiently large qm there exists a primitive normal basis of f m over fq davenport 4 proved in 1968 that a primitive normal basis exists for all m if q is prime in the present paper this result is extended to the general case theorem for every prime power q 1 and every positive integer m there exists a primitive normal basis of fqm over fq section 1 contains an exposition of certain results due to ore 7 concerning the galois module structure of finite fields these lead to an alternative formulation of the theorem in section 2 we describe an improved version of the method of carlitz and davenport which handles all but finitely many pairs q m in section 3 we determine which are the remaining pairs and they are dealt with in section 4 we denote the cardinality of a set s by s and the group of units of a ring r with 1 by r if f g are polynomials in one variable we mean by g i f that g divides f and is monic i e has leading coefficient one the same notation for divisibility is used for positive integers received march 5 1986 1980 mathematics subject classification 1985 revision primary 11t30 12e20

an fft extension to the p 1 factoring algorithm j m pollard in 1974 presented the p integer factoring al gorithm his paper couched the algorithm in theoretical terms based upon use of fast fourier transform techniques but he was unable to say whether the method could be made practical we discuss the mathematical basis of the al gorithm and show how it can work in practice the practical implementation depends for its success upon the use of residue number systems we also present an open problem as to how the method could be made to work for the elliptic curve factoring algorithm

factoring with two large primes the study of integer factoring algorithms and the design of faster factoring algorithms is a subject of great importance in cryptology cf 1 and a constant concern for cryptographers in this paper we present a new technique that proved to be extremely useful not only to achieve a considerable speed up of an older and widely studied factoring algorithm but also and more importantly to make practical application of a new factoring algorithm feasible while this first application does not pose serious threats to factorization based cryptosystems the consequences of the second application could be very encouraging from the cryptanalysts point of view

on orders of optimal normal basis generators in this paper we give some experimental results on the multiplicative orders of optimal normal basis generators in f 2 n over f 2 for n 1200 whenever the complete factorization of 2 n 1 is known our results show that a subclass of optimal normal basis generators always have high multiplicative orders at least o 2 n 1 n and are very often primitive for a given optimal normal basis generator a in f 2 n and an arbitrary integer e we show that e can be computed in o n v e bit operations where v e is the number of 1 s in the binary representation of e