Abstract: Given an integer N, what is the computational complexity of finding all the primes less than N? A modified sieve of Eratosthenes using doubly linked lists yields an algorithm of OA(N) arithmetic complexity. This upper bound is shown to be equivalent to the theoretical lower bound for sieve methods without preprocessing. Use of preprocessing techniques involving space-time and additive-multiplicative tradeoffs reduces this upper bound to OA(N/log logN) and the bit complexity to OB(N logN log log logN). A storage requirement is described using OB(N logN/log logN) bits as well.

Abstract: Let (A,m,k) be a one-dimensional Noetherian local ring of characteristic p (p > 0, a prime number) and assume that the Frobenius endomorphism F of A is finite. Further assume that the field k is algebraically closed and that it is contained in A. Let B denote A when it is regarded as an A-algebra by F. Then, if HomA (B,A) m B as B-modules, A is a Macaulay local ring and r(A) dimk Ext, (k,A) ( max{# Assii-1,1} where A denotes the m-adic completion of A. Thus, in case # AssA 3 this assertion is not true and the counterexamples are given.

Abstract: Let the field K be an abelian extension of the rational field Q. The Schur group of K, S(K), consists of those classes in the Brauer group of K which contain an algebra isomorphic to a simple component of a rational group algebra QG for some finite group G. Suppose that K has a cyclic extension of the form Q(ζ) where ζ is a primitive nth root of unity. In this paper we calculate the 2-part of S(K) where K contains the fourth roots of unity. An interesting facet of these results is that in some cases certain local indices of classes in S(K) are tied together. That is, a class in S(K) must have a nontrivial local index at an even number of the primes in a certaiji set. The tying together of local indices in these fields is caused by quadratic reciprocity and is not found in the g-part of S(K) where q is an odd prime number. Let [A] be the class in the Brauer group of K which contains the iΓ-central simple algebra A. The Hasse invariant of [A] at a prime © of if is denoted inv® [A]. Benard and Schacher [2] showed that each class [A] in S(K) has uniformly distributed invariants. That is, if the index of [A] is /, and σ(sz) = ej where e7 is a primitive Ith root of unity and σ e Gal (K/Q), then inv© [A] = λ invσ(@) [A] for each prime © in K. A corollary of this result is that the local index of a class [A] in S(K) is the same at each of the primes of K which divide a single rational prime p. This common index is called the p-local index of [A], Set L = Q(ξ) where ξ is a primitive 28nth root of unity, (2, n) = 1.

Abstract: Truncatable primes are those that yield a sequence of primes when digits are removed always from the left or always from the right. The sizes of the largest truncatable primes in a given number base are estimated by a probabilistic argument and compared with computed values. The number 357686312646216567629137 is a prime, and, if successive digits are removed from the left, a sequence of primes ending 137, 37, 7 is obtained. We call such a number a left-truncatable prime and this particular one is the largest in decimal notation. Similarly 73939133 is the largest right-truncatable prime to base 10: it yields a sequence of primes . . . 73, 7 if we truncate from the right. The number 1979339339 has been quoted [1] as the largest right-truncatable prime; but we adhere to the convention that 1 is not a prime number, and so exclude it. We have computed La) the largest left-truncatable prime with base a, for 3 < a < 11 , and Ra ,the largest right-truncatable prime with base a, for 3 < a < 15. The results are as follows (in decimal form).

Abstract: In 1933 D. H. Lehmer [3], in connexion with a method for discovering large prime numbers, posed the following question. Let α be an algebraic integer of degree D with conjugates α = α1, α2, ..., αD and put[EQUATION]where a is the leading (positive) coefficient of the minimal polynomial of α.

Abstract: Let p be a prime number, G a finite group whose order is divisible by p, and S a Sylow /»-subgroup of G. We say that G satisfies (•) if there exists g e G such that 51 n Sg « Op(G). There are examples of primes p and finite groups G that do not satisfy (»). In this note we discuss several related properties satisfied in general, as well as giving sufficient conditions for G to satisfy (•).

Abstract: SUMMARY A method is developed for identifying effects and confounding patterns in factorial designs generated by Patterson's (1976) DSIGN algorithm. The method is an extension of that com- monly used for symmetrical designs with prime number of levels. Das (1964) described an equivalent method in which some of the treatment factors are designated as basic factors and the others as added factors. Levels of added factors are derived by combination of the levels of the basic factors over GF(t). White & Hultquist (1965) extended the field method to designs with numbers of levels of treatment factors still prime or prime power but possibly differing from one factor to another. John & Dean (1975) described the construction of a particular class of single replicate block designs, which they call generalized cyclic designs. The levels of treatment factors are now integers 0, 1, ..., t - 1. The essential feature of the method is that the m-tuples giving the treatments of a particular block constitute an Abelian group, the intrablock subgroup. The method gives the same single-replicate designs as the field method when t is prime. Treatment levels are, however, differently represented when t is a power of a prime and so, in general, different designs are obtained. Unlike the field method, the generalized cyclic method is also available when t is neither prime nor prime power. Dean & John (1975) extended their method to asymmetrical designs. Patterson (1976) described a general computer algorithm, called DSIGN, in which levels of treatment factors are derived by linear combinations of levels of plot and block factors. The method provides finite-field, generalized cyclic and other designs. It is not restricted to block designs but is available for any of the simple block structures defined by Nelder (1965) with fractional, single or multiple replication of treatments. In the present paper we are concerned with identification of effects and confounding in the DSIGN method. We tackle the problem by inspecting linear combinations of levels of treatment factors. This approach is, of course, familiar for finite-field designs: Kempthorne (1947) has

Abstract: In [4] we have given a simple method of estimating trigonometrical sums over prime numbers Here we show how the argument can be adapted in order to give estimates for the distribution of αp modulo 1 which are sharper than those obtained by I M Vinogradov [5], [6] Vinogradov uses the sieve of Eratosthenes to relate the sumto the bilinear formthe function μ being the Mobius function When d1 … ds is small compared with N this can be treated in a fairly straightforward manner However, in order to treat the terms with d1 …ds close to N, Vinogradov has to introduce an argument of a rather recondite combinatorial nature

Abstract: Publisher Summary This chapter presents a model of ZF set theory in which the principle of dependent choices and the prime ideal theorem for Boolean algebras are both true while the axiom of choice is false. The problem of finding such a model remained open despite considerable effort. It contains a model that is considered to be the best candidate for the theorem, together with an incomplete proof sketch along the lines of the original independence proof of the axiom of choice from the prime ideal theorem. No progress has been made on this model. The chapter presents some further applications of the method without proof. The prime ideal theorem, Hahn Banach theorem, and canonical uniform ultrafilter principle (a uniform ultrafilter includes all sets whose complements are well orderable and have smaller cardinality. The principle states that there is a function assigning a uniform ultrafilter to the power set of each infinite set) can be added to the class of automatic ZF transferable Fraenkel–Mostowski independences.