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A MATRIX REPRESENTATION OF THE PRIMITIVE RESIDUE CLASSES (mod 2ra) K. MAHLER A problem1 in the geometry of numbers recently lead me to consider some simple matrices with elements 0, 1, and —1. I found to my surprise that these matrices had inverses of the same kind, that they were commutative, and that they in fact formed an Abelian group. These matrices are discussed in the present note. 1. Let m and ra be two positive integers such that 1 < m < ra, Let further i^Obea parameter (m, n) = 1. and / one of its rath roots, t" = s. There are thus » distinct say. possible values for t, the values h, t2, ■ ■ ■ ,t„ Now denote by A(m, ra) = (ahk) and B(m, n) = (bhk) the two raX« matrices with the following elements. For each pair of suffixes h, k = 1, 2, • ■ • , ra determine integers i, j, q, and r such that km — h = i (mod w), 0 < i < ra — 1, q =-, km — h — i ra and km+ h = j (mod n), 1 < j < n, r =-. km + h — j ra Then put ahk = sq if 0 < i < m — 1, ahk = 0 if m < i < ra — 1; hk = s^1 if 1 < j < tn, bhk = 0 if m + 1 < j < ra. Thus, by way of example, Received by the editors March 27, 1956. 1 It has been conjectured that the symmetric convex domain in the plane of given lattice determinant and smallest area is bounded by line segments and osculating hyperbolae arcs. The discussion of such domains leads to systems of linear equations which have matrices just as considered in this note, and I found their group property when I tried to solve the equations. 525 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 526 K. MAHLER 1 0 S2 0 ] 10 5 S2 0 1 0 5 0 52 , 0 1 5 0 52 .0 1 0 s s2) ,4(3, 5) = 5 [June [0 0 1 0 S 0 1 0 5 0 . 1 0 0 5 0 0 5 0 0. 01005 5(2, 5) = (l We shall study these matrices mainly in the case when m is odd and 5 has the value —1, but, for the present, do not yet impose these restrictions. 2. Denote by * = (*a), three variable «X1 y = (yn), matrices (column y = A(m, n)x z = (zi,) vectors) and such that z = B(m, n)x, or in explicit form, n n yn = 23 &hkxk, zh = 23 bnkXk- A-l Further *=1 put, for shortness, Y = yi + Z = t»-hl tyt + t2y3 + + tn~2Z2 + • • • + ■• • + tn~xyn, tZn-l + Then n n n y = 23 23 <*-10**** = 23 ukXk h-l k—l k—l where n itk = 2Z th-lank, h-i and similarly n n n z = 23 23 tn~hbhkxk = 23 »*** A-l *-l *-l where n »* = 23 in~hbhk. n-i License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Zn- i957l PRIMITIVE RESIDUE CLASSES 3. These expressions can be replaced definition of a»* it is evident that r^lh-i^ina+h-i^ikm-i-i t"-lahk=< 527 by simpler ones. From ;f km—h = nq+i, 0<i<m—l, . (.0 the if km—h = nq+i, m<i<n —l. Therefore n m—1 1 _ im uh = 23 <*_1a**= 23 thm-i-x = f(*-i)»-, h—l i=0 1 — t whence n 1 _ y = 23<(i_1)m-** k-i i - im t i — j- =-(xi + On combining Jmx2 + /2mx3 + this formula with ty2 + ■■■ + • • • + i(n-1)mx„). the definition of Y, we obtain the First Identity, (1 ~ t)(yx + = t2y3 + (1 - tm)(Xi + i"-»y») tmXt + t2mx% + • • • + *tn-1)mx„). Similarly, by the definition of bht, {sr-Hn-h=tnr~h=tkm-j tn-hbhk= jf km+k = nr+j, 1</<W, { (.0 if km+h = nr+j, m+l<j<n. Thus now n m h-l )-l 1 _ Vk= 23 'n~^** = 23 '*"*"*' = iC4_1),n-> im I — t hence n J _ ^m z = 23*(fc-1)m-x* *-i 1- i 1 = 1 and therefore, (1 - f)(*-»ii /m -"(*i + /m*2 + i2mx3 + • • • + t<n-1)mx„), t from the definition of Z, + i"-222 + • • • + te„_i + z„) = Here the left-hand (1 - /m)(*i + rmx2 + /2mx3 + side may also be written License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use as • • • + /(n-°m*n). 528 K. MAHLER -t"(l - t-l)(Zl + r1!, + [June t~2Z3 + ■■■ + t-^^hn). Since t" = s, we obtain then the Second Identity, -5(1 - t-^Zi + r% + i-% = tm)(xi (1 - 4. Denote by r an arbitrary + ■■■+ + tmx2 + t-^Zn) t2mx3 + parameter, • • • + l<-"-1)mxn). by * = (60 an arbitrary «X1 matrix (column vector), and put *(£ I r) = (1 - t)(& + r£2 + r2£3 + • • • + r-^). In this notation, the two identities 4>(y | 0 = $(x | tm) respectively. and (1) and (2) take the simple form -s<i>(z \ f1) = *(* | tm), Here, for s^O, t may be any one of h, t2, • ■ ■ , tn. Lemma 1. Let a be distinct from 0 and 1, and let T\, t2, • • ■ , t„ denote the ra roots of the equation t" = <r. For any ra given numbers <pi,4>z,• • • , <Pn there exists one and only one vector £ such that H^\rh)=4>h Proof. (k = 1,2, •••,»). The expression <l>may also be written as $(£ I r») = (£i - cr£„)+ T»tt, - £i) + t'(?3 -&) + ••• + ^o. - ?»-i). The hypothesis a?^0 implies that the ra roots n, r2, • • • , t„ are all distinct, hence that the Vandermonde determinant | Th does not vanish. The assertion that the ra linear forms £l — <r£», £2 — £l, in £i> £2, • • • , £n are linearly of these forms evidently whence the assertion. \h.k-l,2....,n is therefore proved £3 — £*, ' • ' ,r£n independent. However, if it can be shown — £n-l the determinant equals 1 —a and so, by cf^ 1, does not vanish, Lemma 2. Let sm, hence also s and s'1, be distinct from 0 and 1, and let ti, t2, ■ • • , tn be the roots of tn = s. The ra equations (3) Hy\th) = *(x\t7) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use (h = 1,2, •••,»») i957l PRIMITIVE RESIDUE CLASSES define a nonsingular linear mapping 529 of x on y and vice versa; and the n equations (4) -5$(z | h) similarly define a nonsingular = $(* \t?) linear mapping Proof. The assertion is contained <r= s~l, and o-= 5"\ respectively. (h=l,2,---, n) of x on z and vice versa. in Lemma 1 applied with <r= s, Corollary. If sm is distinct from 0 and 1, then the two matrices A(m, n) and B(m, n) are both nonsingular. 5. From now on we impose the additional m is odd, Hence h, tt, • • • , tn now satisfy and conditions that s = — 1. the equation t" = - 1. Thus, while, which x and for odd n, —ti, —tt, • • • , —tn are all the nth roots of unity, for even n, h, t2, ■ ■ ■ , t„ are all those (2«)th roots of unity are not also nth roots of unity. The equations (3) connecting y remain unchanged, but the equations (4) between x and z now become or equivalent Hz | h1) = $(* | h) (h - 1,2, •••,»), *(i | *,)-*(* I O (h = 1,2, •••,«). to this, (5) Since, by hypothesis, m is prime to n, and further obvious that both the mtki powers m is odd, it is ti , h , * • • , tn, and the ( —m)th powers 'i , h , • • • , tH of h, tt, • • • , tn are again these same roots, only possibly arranged in a different order. For, first, t"= — 1 implies that also (tm)n = (t~m)n = —1 because m is odd. Secondly, by (m, n) = l, there exist integers M and N such that mM+nN=l. (fm Hence, if tn = t'n= -1 and t^t', \ M / (n\ N -J t'V ( —) \t'n/ then l = — ^ 1 and therefore lm ^ t'm, t~m 7* t'~m. t' License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 530 K. MAHLER [June 6. From now on we change the notation a positive or negative integer such that (6) (m, n) = 1, In extension slightly 1 < | m | < ra — 1, of the previous notation and allow m to be m is odd. we then put CA(m,n) if m > 0, A(m, ra) = < (B(—m, n) if m < 0. Therefore, in either case, the mapping y = A(m, n)x is equivalent to the system of ra formulae, Hy\h) = Hx\h) (h = 1, 2, • • • , ra). Next, let m' be a second integer satisfying the conditions case when m'=m is not excluded. Further let (6); the z = A(m', n)y so that also z = A(m', n)A(m, n)x. The definition of z implies that 4>(z| th) = *(y\t?) Now, as we saw above, (* = 1,2, • ■ • , ra). we can write m' th — tkih) where / 1 U(l) is a certain permutation. 2 • •■ ra \ k(2) ■■■k(n)J Therefore $(z | th) = Hy | h ) = <K;y| <*w) = $(a; | hw) and finally *(a|/») = Hx\ihn'm) (h = 1, 2, • • • ,«). By the hypothesis, mm' is odd and prime to ra, hence also prime to 2ra. Hence there exists a unique integer p. such that License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use i957l PRIMITIVE RESIDUE CLASSES fi = m'm (mod 2ra), and therefore 531 1 < |p\ < n —1 also (p, n) = 1, The congruence p. is odd. for p implies in particular m'm that p k = h (h = 1, 2, • • • , ra). It follows then that 9(z\th) = Hx\fh) These equations show, however, that (h = 1, 2, • • • , ra). necessarily z = A(p, n)x and we obtain the final result that A(m', n)A(m, n) = A(p, n). The following Theorem. theorem has thus been proved. The <p{2n) matrices A(m, ra), where (m, 2n) = 1, 1 < \m\ < n — 1, form under multiplication an Abelian group which is isomorphic to the group of primitive residue classes (mod 2ra). The isomorphism is defined by A (m, n) <-> {m (mod 2ra)}. Manchester University, Manchester, England License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use