On the Arithmetic of Quaternions
Gordon Pall · Transactions of the American Mathematical Society · 1940
PALL 1.Some fifteen references, beginning with Euler, abstracted in Dickson's History,] use a connection between the congruence and equation 2 2 2 2 2 2 2 1) vi + v2 + v3 = 0 (mod m), t0 + h + h + h = m, in the case where m is a prime, to prove that every positive integer is a sum of four squares.This connection is made precise, for arbitrary odd m, in Theorem 4.Among our theorems will be found conditions for quaternions to have the same right or left divisors of given norms.An easy derivation of known relations concerning binary quadratic class-numbers and representation in x\+x%-{-x\ concludes the article.Most of these results were discovered during an investigation of the "rational automorphs of x\-\-x22-\-x\," and many of them are used in an associated article of that name.Since these automorphs are connected more simply with Lipschitz quaternions than with those of Hurwitz in which the coordinates may be halves of odds, our results are stated for the former type of integral quaternion, although there is no difficulty in extending them to the latter.No further mention is made in this article of automorphs.% Notations.The letters a,b,c,t,u, ■ ■ ■ ,z denote integral quaternions of the type / = 4+^1+4/2+^3, with rational integer coordinates U. Conjugate and norm are defined as usual: t -ta-X/X, Nt = it='^lt2f.Except for the quaternion units ia, letters with subscripts, and d, e, ■ ■ , s denote rational integers; p denotes an odd prime, m an odd positive integer.Subscripts a or ß range over 1, 2, 3; i, j,f, or g over 0, 1, 2, 3.We call / pure (mod m) if m \ t0; proper (mod m) if the g.c.d. of to, ■ ■ ■ , t3, and m is 1; proper if the g.c.d. of to, ■ • ■ , t3 is 1.German capitals represent the sets defined as follows:* Presented to the Society in part April 15, 1933, under the title On the relations between sums of three and four squares, and in part April 13,1940; received by the editors January 31,1940.This paper was received by the editors of the Annals of Mathematics July 5, 1939, accepted by them, and later transferred to these Transactions.t L. E. Dickson, History of the Theory of Numbers, vol.2, chap.8. X The complications of Lipschitz's article are partly due to their use.(R. Lipschitz, Journal de Mathematique, (4), vol. 2 (1886), pp.373-439.)Several papers by the writer, now published, owe their inception to the study of the rational automorphs, and some of their proofs now stated in terms of quaternions were originally obtained by automorphs.The associated article mentioned above will appear soon in the Annals of Mathematics.