H. S. Vandiver

Note on a simple type of algebra in which the cancellation law of addition does not hold

converge to a function </>(V) analytic in 5 and such that | <f)(z) | <1 for z in S. Since the original sequence {pn(z)} converges by hypothesis in R, the limit function of the subsequence would necessarily agree with ƒ(z) in R. But <j>(z) could not be identical with f(z) in R} for then <j>(z) would have to equal unity at interior points of S, namely, at the boundary points of R which lie interior to S. It would appear from this remark that Theorem 1 (and Theorem 2 as well) might admit of extension to an arbitrary finite simply connected region whose complete boundary is also the boundary of an infinite region, and that a modification of the methods used in the proofs of the present paper would serve to establish such an extension. The writer hopes to answer this question in a later paper.

On Fermat’s last theorem

In two recent paperst the writer stated without proof four theorems concerning Fermats last theorem. These results were employed to prove the theorem for all exponents greater than 2 and less than 211. In the present paper the proofs of Theorems I to IV of my previous papers will be given, as well as one of an additional Theorem V. I shall first indicate the relation of the present results to the previous work done on the problem. In the year 1850 Kummert proved that the equation

Fermat’s last theorem and the second factor in the cyclotomic class number

where / is an odd prime and x, y, z are rational integers prime to each other and none zero, we shall refer to the case where xyz is prime to / as case I ; if xyz = 0 (mod I) then we call this case II . I now give a sketch of a proof of a theorem which appears to be the principal result I have so far found concerning the first case of the last theorem. THEOREM 1. If (1) is possible in case I, then the second factor of the class number of the cyclotomic field defined by

On the Imbedding of One Semi-Group in Another, with Application to Semi-Rings

In other papers 1 a semi-group was defined as a set of elements closed ander an associative operation and for which the equivalence and the substitution postulates hold. tn the present paper we shall employ instead of the substitution postulate, the postulate that if A = B, then CA = CB and AC = BC for any A, B or C in the set, which we shall call the composition postulate.2 A gruppoid 3 is a semi-group with an identity element, that is, an E such that, AE = EA = A for any A in the set. A quasi-group is a semi-group such that from either of the relations

Note on Fermat’s last theorem

for each factor r of x, in case x + O ( mod p ), and for each factor r of X2 _ y2 s in case X2 _ y2 iS prime to p. By applying this theorem, Furtwangler deduces the criterion of Wieferich q (2) 0 (mod p) and the criterion of Mirimanoff q (3) _ O (mod p) for the solution of (1) in integers prime to p. I shall here extend these results and show that in addition we have, provided that q (2) + O (mod p3 ), the criteria q (5)-O (mod p) for p _ 1 (mod 3) and q (5)-q (7)-O (mod p) for p = 2 (mod 3). 2. Assume that x, y and z are prime to each other and to p and that p > 5 . If one of the integers x, y, z is divisible by 5, then q (5)-O (mod p) by Furtwanglers theorem. If none of them is so divisible, then, modulo 5, xP,yP,zPhavetheresidues 2, 2, 1 or 1, 1, =F2insomeorder.

Theory of finite algebras

1. The object of this paper is to develop an abstract theory of finite algebras which is applicable to various familiar and important concrete algebras, such as the m classes of residues of integers modulo to, the classes of residues of polynomials in x with respect to the moduli m and P (x), the classes of residues of integral algebraic numbers of a given algebraic field with respect to any ideal as modulus, and the classes of residues with respect to certain modular systems. We consider an algebra composed of a finite set of elements which may be combined by addition, subtraction and multiplication, subject to the commutative, associative and distributive laws, and such that the sum, difference or product of any two elements is uniquely determined as an element of the set, while, moreover, there occurs an element playing the rôle of unity under multiplication. It is not assumed that division is always possible; a product may vanish when neither factor vanishes. While the field R (ai, ■ ■ •, an ) defined by the algebraic numbers ai, • • •, a« is identical with a field R (u) defined by a single algebraic number u, a similar theorem does not hold for finite algebras (§6). As regards their units, primes, etc., the theory of finite algebras presents analogies with the theory of integral algebraic numbers. I am indebted to Professor L. E. Dickson for valuable suggestions. 2. Definition of a Finite Algebra. Let the elements u0, Ui, • • •, m,_i form a commutative group under addition. The unique element «o such that u,; 4M0 = u,; ( i = 0, • • • , s — 1 ) is called the zero element. It is assumed that the elements may be combined by multiplication, subject to the commutative, associative and distributive laws, and that the product of any two elements is an element of the set. We shall discard the assumption, made in the theory of finite fields, that division by every element other than u0 is possible and unique. However, we shall assume that there exists at least one element m* , called a unit element, such that Uk x = u0 has the unique solution x = «o • Elements which are not units are called non-units and denoted by Ni,Nt, •••• Theorem I. Division by a unit Uk is always possible and unique. Since the products Ui Ut (i = 0, 1, •••,*— 1 ) are distinct, they form a permutation of the Ui. Thus xllk = ui has one and but one solution x.

On the Computation of the Number of Solutions of Certain Trinomial Congruences

In this article the work is confined to the consideration of the congruence, with p an odd prime, (1) where cl + 1 = p, and xy ~ 0 (mod p). For small values of 1 and p, (1) was considered by E. H. Pearson [1], and a tmmber of results were tabulated (p. 1284), based on using a primitive root of p, say g, and studying the values of r and s in the congruence 1 + g~+Cr ~ g~,Z, (mod p), (2) where i and j are fixed, 0 ~ i =< c-1; 0 ~ j =< l-1. By the properties of primitive roots the problem of finding the incongruent solutions x, y; xy ~ 0 (mod p) which satisfy congruence (1) reduces to finding the distinct sets r, s which satisfy equation (2), ~ith r in the set 0, 1,-.. , 1-1 and s in the set 0, 1,. .. , c-1. We shall denote the number of solutions r, s in the relation (2), within the limits mentioned, by [i, j]cz ; or [i, j] if c and 1 are fixed during an argument. In another article [2] it was shown that a relation between the problem just mentioned and the Fermat problem existed. In 1954 Emma Lehmer coded the general problem of calculating the values of the [i, j]s on the SwAc and checked all the previous tables [1] giving values of the [i, j]s. In the spring of 1955 Nico] and Selfridge extended this program on the Sw.~c and tabulated all the values [i, j] for l = 5, for each p-= 1 (rood 5) for p < 1024; also for 1 = 7 with all ps < 1024; p ~ 1 (rood 7). For 1 = 11 all the [i, j]s were tabulated for each p < 800; p ~ 1 (mod 11). Similarly, for each p < 600 with 1 = 13 the values were determined. Also similar determina-tions were made for each p < 512: p =-1 (mod l) where each 1 is taken in the set 17 =< l < 256. The following results are known [3]:

On the Desirability of Publishing Classified Bibliographies of the Mathematics Literature

Having been engaged now in mathematical research for 60 years, I find that certain crying needs stand out very clearly as to the things required by research mathematicians in order to produce and publish more articles, particularly original ones. To my mind, first among these needs is the preparation and publication of bibliographies classified according to topics of all branches in pure and applied mathematics. Accordingly, this paper will be devoted to this topic and will be divided into two parts. The first part will consist of a discussion of a bibliography of the Theory of Numbers either with reports, as a continuation of Dicksons History of the Theory of Numbers, or a preparation of a bibliography consisting of titles only without reports, covering the literature since the year 1923. The second part will consist mainly of consideration of the advisability of publishing bibliographies of the various parts of mathematics without reviews. First, we shall illustrate the kind of thing which mars a number of papers in number theory that appear at the present time, and probably we might note a great many more marred in a similar way if we knew more about the history of the topic being discussed. Some years ago two great mathematicians (whom we shall call X and Y) published an article. In their introduction they summarized, without reference, two results as being the principal ones being proved in their paper. One of these results was published without proof by a mathematician some 90 years before. The other result was stated and proved in full some 20 years before, by another mathematician. If I had written X and Y, or published a paper describing their derelictions, they could have replied that a paper of mine, which I published in 1944, contained a result and proof which was 109 years old. How can such duplications be avoided? A number of mathematicians have expressed the hope that Dicksons History be brought up to date. Before doing so, it might be well to examine in detail some of the salient features of the book with a view to preserving them in the proposed extension. As far as I am concerned, I cannot do better than to quote something by Dickson himself (preface to Volume II):

On the distribution of quadratic and higher residues

Since by hypothesis H(t,t)^0, we conclude that the derivative of fp+i(t) exists and is continuous and that ^ + i ( l ) = fp±i(0) — 0. Hence it is necessary that the linear functional of continuity order p reduce, by an integration by parts, to a linear functional of continuity order p—1. The lemma applied again to this new functional reduces it to a linear functional of continuity order p—2. Applying the lemma successively in such a manner, we finally get, as a necessary condition for invariance, the result that the original linear functional of continuity orders has to reduce at least to a linear functional of continuity order one. This result coupled with the existence theorems cited in the beginning of § 5, establishes our theorem.

Volumes II and III of Dickson's History

as H. J. S. Smith, and Vandivers numerous contributions to the same end. Last there is the recent work of Dickson, which bids fair to be epoch making, on Algebras and their Arithmetics,* where the classic theory of algebraic numbers finds a simple and profound generalization. With the books of Landau and Dickson, the report of Hubert and that of the present authors now available, it is to be hoped that algebraic numbers, one of the major divisions of modern mathematics, will not much longer remain in learned obscurity, but will take its rightful place as one of the chief glories of any liberal mathematical education.