ON A CONSTRUCTION FOR THE REPRESENTATION OF A POSITIVE INTEGER AS THE SUM OF FOUR SQUARES

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1 ON A CONSTRUCTION FOR THE REPRESENTATION OF A POSITIVE INTEGER AS THE SUM OF FOUR SQUARES Autor(en): Rousseau, G. Objekttyp: Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 33 (1987) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Persistenter Link: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 ON A CONSTRUCTION FOR THE REPRESENTATION OF A POSITIVE INTEGER AS THE SUM OF FOUR SQUARES by G. Rousseau Hermite [4] gave a very simple construction for the représentation of a prime of the form 4/c + 1 as the sum of two squares, using continuée! fractions. This note is concerned with the extension of this construction to the domain of Gaussian integers. In 1 we show that, as might be expected (although it does not seem to hâve been published previously), there is a similar construction in the domain of Gaussian integers for the représentation of an arbitrary positive integer as the sum of four squares. One might also expect some small additional complications in the four squares case, and indeed what is shown in 1 is that for a given positive integer m there is a représentation of m, 2m or 3m, from which one then passes to a représentation of m. This is of course not difficult, but it is somewhat inélégant. However, in computations carried out by L. Rousseau and the author, it emerged that this subsidiary transformation is never in fact needed. The construction always yields a représentation of m directly ; in other words, the Hermite construction applies just as well for the construction of four squares représentations as it does for two squares représentations. This is proved in 2 by a somewhat more elaborate argument, using an old resuit of Auric [1] on continued fractions with Gaussian integer terms. Some of the other known four squares constructions are discussed briefly in We consider the extension of Hermite's construction to the domain of Gaussian integers. Any positive integer N may be expressed in the form N= k 2 m, where mis square-free. In order to construct a représentation for N as the sum of four squares, it is evidently sufficient to construct one for m. The congruence = 0 (mod m) is solvable in rational a2a integers, so, setting a integers, = a + b2b ib, we hâve, in the domain Z + TA of Gaussian

3 (1) Performing the Euclidean algorithm in Z + Zi, beginning with y0 = a, y l = m, we obtain Gaussian integers K r, y r +2 sucn tnat terminating with yn +2 = 0. Thus a/m has the continuée fraction development If [k s,..., kj dénotes the usual Gaussian bracket, then a r = [k 0,kx.., kj and (3 r = [k x.., kj are, respectively, the numerator and denomi natorof the rth convergent of the continued fraction for a/m. We hâve y r =ym+ 1 [K r,..., K n ], where y n +1= (a, m) is a unit. Also the following standard formulas may be established : (2) (3) From (3) we hâve ap,. = (? l) r y r +2 (mod m), so that, in view of (1), Thus, for any r, N(fir ) + N(y r +2) cc =aï+b?+ 2 r + àà r 2, say, is a multiple of m. We shall show that there exists r such that To this end we fîrst show that This is done by induction on r. Thus let w r = (Vy,. + i/yi; it is to be \ shown that \w r <2. w r =1- pp Clearly w 0 = P o = 1- In view of (2) we hâve r -iyr+2/yi =1- (Yr + 2/Yr)w r-i; since, for r>o, yr+2 /Yr I y, we hâve wr < 1 +- w,_ x\, so if ww r _ x <2 then wr \ <2,as required. Now, since the N(y r +1) decrease monotonically from m2m2 to zéro, we may (if m>l) choose r so that N(y r+2 ) < 2m < JV(y ). r+l Then, on the one hand, +2)< 2m > wnile > on the other hand, JV(P ) r < 4m 2 /AT(y r +1) < 2m. Thus 0< N(fi r ) + N(y r +2) < 4m, as was to be shown.

4 Thus m, 2m or 3m is the sum of four squares, from which it easil} follows that m is representable as the sum of four squares (cf. [2]). Indeed if 2m a2a = 2 b2b + 2 c2c then by rearranging the terms if necessary w( d2d may suppose a = b (mod 2), c = d (mod 2), so that where the terms on the right are integers, while if 3m = a2a 2 + b2b 2 + c2c 2 + then by changing signs if necessary we may suppose a, b7b 7 c, d = 0 or 1 (mod 3) and further, by rearranging the terms if necessary, that a = 0 (mod 3) b = c = d (mod 3), so that where again the terms on the right are integers 2. We shall establish the following resuit Theorem. Let m be a square-free positive integer and let aot =?1 (mod m). // a/m has the continued fraction development then for some r n we hâve where Thus a représentation of m as the sum of four squares is obtained by performing the Euclidean algorithm for a/m, calculating p r at each step, until an ris reached for which MB_) + My p +9) = m. If ail prime divisors of m are of the form 4/c+l then a and the K r, y r may be taken as rational integers and so the construction reduces essentially to that of Hermite [4]. Proof For any r, N(pr ) + N(y r +2) =a r +c2 + dd r 2 is a multiple of m. We must show that for some r the multiple of m in question is m itself. It is clear that N((3 r ) + N(y r+2 ) >0, since if N{y r+2 ) =0 then r=n and so MB r ) = MBJ =m2 >0.

5 I Auric [1] showed that (4) and that (5) If IYoYII< 3m/2 then it is easy to see that m 3 and N(y 0 ) + N(y 2 ) =m. In the contrary case, the y r y r +il decrease monotonically from y0 Yi 3m/2 to zéro. Hence we may choose r so that (6) For each r let N r = N($ r ) + iv(y r + 2); then N r = k r m for some positive integer k r. By Euler's identity and (2) we hâve Now by (5) and (6), and by (6). If 3m/2 < yr+ly r+1 2 then by(s). If y,+ il 2 < 3m/2 then Also by (4) and (5). Hence we hâve

6 It follows that K-iK = 1, 2, 3, 4, or 5. But if /c rr _ x /c r = 4 then 3m - 2 pp r _ip r - Yr-iYr^il 2. so that 3m 2 is the sum of two rational intégral squares, which is impossible. Hence kk r _ x k r is a factors is unity, i.e., A^p^) + N{y r +1) =m or JV(P r ) prime or unity and so one of the + N{y r +2) = m. This complètes the proof. The above argument shows that for any a (not necessarily satisfying (1)) there exists r such that Computation shows that for m 200 and arbitrary a there exists r such that It would be of interest to know if the same is true for ail m. 3. Among the other known constructions for the représentation of a positive integer as the sum of four squares is that provided by the réduction theory of positive definite Hermitian forms ([s]). Equivalent to this is Smith's construction [6], based on representing the fraction m/a as a conjugate symmetriccontinued fraction (7) which gives, in view of (2), Another construction is known from the theory of intégral quaternions. Namely, if (1) holds then the (right) g.c.d. q = (m, a+j) exists and m = N(q) (hère we may interpret "intégral" either in the sensé of Lipschitz or of Hurwitz; in either case q may be taken as having rational intégral coefficients). This may be seen from Smith's construction; thus, with the notation as in (7), set then we hâve from which Le.,

7 If only the existence of a représentation is required, rather than an explicit construction, then, as is the case in other contexts, the use of continuée! fractions may be replacée by the use of Dirichlet's pigeonhole principle or results in the geometry of numbers which follow from it (cf. Brauer and Reynolds [2], Davenport [3]). Indeed the existence of a solution of oc, = r\ (mod m) with 0 < N(<Q + N(r\) < 4m is a conséquence of Minkowski's theorem on linear forms, while the existence of a solution with 0 < N(Q + N(r\) < 2m follows from Minkowski's theorem on convex bodies. The author is indebted to L. Rousseau for assistance with computational work in relation to this paper. REFERENCES [1] Auric, A. Essai sur la théorie des fractions continues. J. Math, pures et appl. (5), 8 (1902), [2] Brauer, A. and T. L. Reynolds. On a Theorem of Aubry-Thue. Canadian J. Math. 3 (1951), [3] Davenport, H. The Geometry of Numbers. Math. Gazette 31 (1947), [4] Hermite, C. Note sur un théorème relatif aux nombres entiers.?uvres 7, Gauthier-Villars, Paris (1905), 264. [5] Sur la théorie des formes quadratiques, 2.?uvres /, Gauthier-Villars, Paris (1905), [6] Smith, H. J. S. De fractionibus quibusdam continuis. Collected Math. Papers, 11, 0.U.P., Oxford (1894), (Reçu le 10 avril 1987) G. Rousseau Mathematics Department The University. Leicester LEI 7RH (England)