Abstract: (1971). On the Prime Divisors of Polynomials. The American Mathematical Monthly: Vol. 78, No. 3, pp. 250-266.

Abstract: This paper is an elementary note which indicates how Harrison's primes sit in certain kinds of rings. It is proved that primes behave nicely under finite direct products. Also it is shown that any nil ideal is a subset of every prime. This gives information about the primes of artinian rings.

Abstract: A "prime" in an arbitrary ring with identity, as defined by D. K. Harrison, is shown to be a generalization of certain objects occurring in the classical arithmetic of a central simple i£-algebra 2, i.e., the theory of maximal orders over Dedekind domains with quotient field K. Specifically, if K is a global field the "finite primes" of 2 (in Harrison's sense) which contain a iΓ-basis for 2 are the generators of the Brandt Groupoids of normal ^-lattices, R ranging over the nontrivial valuation rings of K. The situation when J contains a finite prime invariant under all i£-automorphisms is studied closely; when K is the rational numbers or char (IT) Φ 0, and 2 has prime power degree, such a prime exists if and only if J is a division algebra. The techniques developed here are applied to yield new information concerning the generators and factorization in the Brandt Groupoids over certain Dedekind domains.

Abstract: Let SF = SG denote the space , and BSF be the classifying space of SF. Our purpose is to determine H *(BSF: Zp) as a Hopf algebra over Zp where p is an odd prime number. We have announced the main result in [14].

Abstract: One of the corollaries of the fundamental theorem in this paper is a theorem about power residues and nonresidues mod q in sequences of the form p + k, where the prime numbers p belong to the beginning of an arithmetic progression.

Abstract: This paper generalizes and extends the recent results of George Andrews on Euler pairs. If Si and S2 are nonempty sets of natural numbers, we define (S1, S2) to be an Euler pair of order r whenever q7(Si; n) =p(S2; n) for all natural numbers n, where q7(Si; n) denotes the number of partitions of n into parts taken from S1, no part repeated more than r -1 times (r > 1), and p(S2; n) the number of partitions of n into parts taken from S2. Using a method different from Andrews', we characterize all such pairs, and consider various applications as well as an extension to vector partitions.

Abstract: Editor's note: The two algorithms published here are available on magnetic tape at a charge of $12.00 i f we supply the tape, or $6.00 if the requester supplies the tape. Checks are to be made payable to \" A C M Algorithms\" and are to be sent with the order to the department editor at his address. The two algorithms are recorded as one file each of BCD 80 character card images at 556 B.P.I., even parity, on seven track tape. The cards for each algorithm are sequenced starting at 1 and incremented by 1. The sequence number is right justified in column 80. Although we will make every attempt to insure that the algorithms on the tape faithfully duplicate the algorithms printed here, we cannot guarantee it, nor can we guarantee that the algorithm is correct. This offering is an experiment. I f there is a good response, we hope to be able to make similar offerings in the future. We welcome your comments and suggestions on this new feature o f the Algorithms department.-L.D.F.

Abstract: Factorizations of various functions are discussed. Complete factorizations of certain classes of functions are given. In particular it is shown that there exist primes of arbitrary growth.

Abstract: Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.

Abstract: The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G , must there be a minimal set of generators of G that contains an element whose order is divisible by p ? A set of generators of G is minimal if no set with fewer elements generates G . A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G ; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powell's question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

Abstract: Let BSPL be the classifying space of the stable oriented PL micro bundles. The purpose of this paper is to determine H*(BSPL:Zp) as a Hopf algebra over Zp , where p is an odd prime number. In this chapter, p is always an odd prime number.

Abstract: In algebraic manipulation one often wants to know if two expressions are equivalent under the algebraic and trigonometric identities. One way to check this is to substitute a random value for each variable in the expressions and then see if both expressions evaluate to the same result. Round-off and overflow errors can be avoided if the evaluation is done modulo a large prime number. Of course, there is still a small probability of a random match. If arithmetic expressions in exponents are evaluated, using the same prime number then identities such as xl/2 × xl/2 = x will not necessarily hold. However, by proper choice of the prime number and the use of special case checks these identities as well as many trigonometric identities can often be preserved.

Abstract: Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value theorems.- Large moduli theorems.- Further results and conjectures concerning mean and large moduli.- Mean moduli of L-functions.- Zero-free regions and the proliferation of zeros.- Distribution of zeros of L-functions.- Least character non-residues and arg L(12+it, x).- The prime number theorems of Hoheisel and Selberg.- The bombieri - Vinogradov theorem.- A lemma in additive prime number theory.- The mean value theorem of Barban.

Abstract: (1.1) D(Q; x,h)-, max] ,(x+h; q,/)--,(x q, 1)-h--h[ qQ (,q)=l. q(q) .’ where (x;q, l)is the usual ebyev function for the arithmetic progression l mod q and Q, h are appropriate functions of x. Both results of Bombieri and Jutila have been obtained by reducing the problem to the estimation of the total density of zeros of ’many’ L-functions. Bu Gallagher [2] has found a way to prove Bombieri’s result without using the density theorem. In [4] an opinion is expressed that it seems dicult to prove a non-trivial estimation of (1.1) on the similar line. The purpose o the present paper is to offer such a proof. Our main tool is the following beautiful inequality of Gallagher [3]: If E nan +,

Abstract: We determine all the symmetric graphs with a prime number of vertices. We also determine the structure of their groups.

Abstract: In this paper the Hughes conjecture for finite pgroups is proved in the case of groups which are nilpotent of class 2p-2. If G is a group and p is a prime number then the Hughes subgroup HpiG) of G is by definition the subgroup generated by all elements of G that do not have order p. The conjecture of Hughes was that if G>HpiG)>l then HviG) must have index p in G; see [o]. Hughes showed that the conjecture is true for p = 2 and all G, and Straus and Szekeres [lO] proved it for p = 3 and all G. Hughes and Thompson [7] proved the corresponding result for all p and for G finite and nonnilpotent. The most interesting cases which remain are those in which G is a finite 7^-group with p ^ 5 ; throughout this note we shall take G to be a finite £-group, and our results will be of most significance when pit5. We may remark at this point that it is a nontrival exercise to construct a £-group GP for each prime p such that Gp>HPiGp) > 1. Though Wall [ll] has shown by means of an example that the Hughes conjecture is false for finite 5-groups and p — 5,it is nevertheless of some interest to find sufficient conditions for the conjecture to hold. Thus Zappa [12] has shown that all is well when G is nilpotent of class p. The main result of the present note is a substantial strengthening of this: Theorem. The Hughes conjecture is valid for finite p-groups of nilpotency class 2p — 2. It seems likely to us that class 2p — 2 cannot be replaced by a much greater class without falsifying the theorem. The proof of this theorem depends heavily on the main result of [8], which we state again in the interests of clarity: Theorem [8]. If \G:HPiG)\^p2, V„(G)£t(G) and y2p(G) = l, thenVPiHpiG)r\y2iG)) = l. Received by the editors February 3, 1970. AMS 1970 subject classifications. Primary 20D15, 20D25.

Abstract: Some results in number theory, including the Prime Number Theorem, can be obtained by assuming a random distribution of prime numbers. In addition, conjectural formulae, such as Cherwell's for the density of prime pairs (p, p + 2) obtained in this way, have been found to agree well with the available evidence. Recently, primes have been determined over ranges of 150,000 numbers with starting points up to 1015. Statistical arguments are used to obtain a formula for the largest interval between consecutive primes in such a range, and it is found to agree well with recorded values. The same method is applied to predict the maximum interval between consecutive primes occurring below a given integer. Introduction. Jones, Lal and Blundon [1] have tabled the largest interval between primes in each of the regions (x, x + A) with A = 150,000, x = 10, n = 8(1)15, as well as the numbers of prime pairs (p, p + 2), etc., lying in these ranges. They found that these counts of pairs, triples, etc. agree well with those given by the conjectured formulae of Hardy and Littlewood [2]. It is interesting to note that Cherwell [3] obtained the same formulae using statistical methods. This note assumes that primes occur randomly in a specified region with a uniform distribution function. The largest interval between consecutive primes in the range (x, x + A) is investigated; for large x its mean value is (1) log x log(A/log x), but for the ranges in [1] it pays to use the rather more elaborate result found below, and agreement with actual counts is good. From Table 1 of [5], we can obtain values of the maximum interval between consecutive primes for the ranges (10m, 1 0 ), m = 3 (1)9, and again the calculated maxima are in good agreement. The method can be used to estimate G(N), the maximum interval below N. The result is G(N) -log N(log N log log N). Received December 7, 1970, revised March 25, 1971. AMS 1969 subject classifications. Primary 1042.