BINOMIAL COEFFICIENTS AND JACOBI SUMS

Transcription

1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 281. Number 2, February 1984 BINOMIAL COEFFICIENTS AND JACOBI SUMS BY RICHARD H. HUDSON AND KENNETH S. WILLIAMS1 Abstract. Throughout this paper e denotes an integer > 3 and p a prime = I (mod e). With/ defined by p = ef + 1 and for integers r and s satisfying 1 *s s < r =c e 1, certain binomial coefficients (r ) have been determined in terms of the parameters in various binary and quaternary quadratic forms by, for example, Gauss [13], Jacobi [19,20], Stern [37-40], Lehmer [23] and Whiteman [42,45,46]. In 2 we determine for each e the exact number of binomial coefficients (JÍ) not trivially congruent to one another by elementary properties of number theory and call these representative binomial coefficients. A representative binomial coefficient is said to be of order e if and only if (r, j) = 1. In 3-4, we show how the Davenport-Hasse relation [7], in a form given by Yamamoto [SO], leads to determinations of n(p~*)/m in terms of binomial coefficients modulo p = ef + 1 = mnf + 1. These results are of some interest in themselves and are used extensively in later sections of the paper. Making use of Theorem 5.1 relating Jacobi sums and binomial coefficients, which was first obtained in a slightly different form by Whiteman [45], we systematically investigate in 6-21 all representative binomial coefficients of orders e = 3,4,6,7,8,9,11, 12, 14, 15, 16, 20 and 24, which we are able to determine explicitly in terms of the parameters in well-known binary quadratic forms, and all representative binomial coefficients of orders e = 5,10, 13, 15, 16 and 20, which we are able to explicitly determine in terms of quaternary quadratic decompositions of 16/7 given by Dickson [9], Zee [51] and Guidici, Muskat and Robinson [14]. Some of these results have been obtained by previous authors and many new ones are included. For e = 1 and 14 we are unable to explicitly determine representative binomial coefficients in terms of the six variable quadratic decomposition of 72 p given by Dickson [9] for reasons given in 10, but we are able to express these binomial coefficients in terms of the parameter x, in this system in analogy to a recent result of Rajwade [34], Finally, although a relatively rare occurrence for small e, it is possible for representative binomial coefficients of order e to be congruent to one another (mod p). Representative binomial coefficients which are congruent to ± 1 times at least one other representative for all p = ef + 1 are called Cauchy-Whiteman type binomial coefficients for reasons given in [17] and 21. All congruences between such binomial coefficients are carefully examined and proved (with the sign ambiguity removed in each case) for all values of e considered. When e = 24 there are 48 representative binomial coefficients, including those of lower order, and it is shown in 21 that an astonishing 43 of these are Cauchy-Whiteman type binomial coefficients. It is of particular interest that the sign ambiguity in many of these congruences does not arise from any expression of the form n(p~1)/m in contrast to the case for all e < 24. Received by the editors February 8, Mathematics Subject Classification. Primary 10-02, 10C05, 10B35; Secondary 12C25. Key words and phrases. Binomial coefficients (mod p) in terms of binary and quaternary quadratic forms, Jacobi sums. 'Research suported by Natural Sciences and Engineering Research Council Canada Grant No. A American Mathematical Society /84 $ $.25 per page

2 432 R H. HUDSON AND K. S. WILLIAMS 1. Introduction and summary. Throughout this paper e denotes an integer > 3 and p a prime = 1 (mod e). The integer/ is defined by p = ef + 1. For integers r and í satifying 1 < s < r < e 1, certain binomial coefficients (1.1) \sf have been determined modulo p by, for example, Gauss [13], Jacobi [20], Stern [37-40], Lehmer [23], and Whiteman [42,45,46] in terms of representations of p by certain quadratic forms. The first result of this kind is due to Gauss [13, Vol. 2, p. 90] who showed that for e - 4, p = 4/ + 1 = a2 + b2, a = 1 (mod4), (1.2) (y) = 2a (modp). Emma Lehmer [23] used Jacobsthal sums to obtain congruences for (2j) and (3{) when p = 5f + 1 in terms of the system I6p = x2 + 50m2 + 50u w2, xw = v2 Auv u2, x = 1 (mod 5), introduced by Dickson [8]. In this paper we systematically use Jacobi sums to obtain congruences modulo primes p = ef + 1 for binomial coefficients of the type (1.1). These include old as well as new ones. The cases treated in 6-19 are e = 3,4,5,..., 16. In we handle in some detail the cases e 20 and e 24, relying heavily on recent evaluations by, e.g., Berndt and Evans [4] of bidecic and biduodecic Jacobi sums. Our results are obtained in terms of the parameters in the following Diophantine systems: (1) p = a2 + b2, a = l(mod4), 4 <?, (2) p = x2 + 3y2, jt = l(mod3), 3 <?, (3) 4p=A2 + 2W2,,4 = 1 (mod 3), 3 e, (4) 16/7 = x2 + 50u2 + 50t) w2, xw = v2-4uv - u2, (5) p = x2 + 7v2, x = 1 (mod 7), e = l, 14, x = 1 (mod 5), (6) lip = 2x2 + 42(x + xj + x ) + 343(x52 + 3x ), jc, = 1 (mod 7), (7) p = c2 + 2d2, c = l(mod4), e = 8,16,24, (8) \6p = x2 + 26w2 + 26ü2 + 13w2, xw = 3v2 - Auv - 3u2, (9) p = g2+\5h2, g=l(mod3), e=\5, 5\e, e = 7,14, jc = 9(modl3), e= 13, (10) p = x2 + 2u2 + 2v2 + 2w2, 2xv = u2-2uw-w2, x = 1 (mod 8), u = v = w = 0 (mod 2), e = 16, (11) p = e2 + 5f2, É> = a(mod5)if5 > and e =\b\ (mod5) if 5 a, e = 20,

3 BINOMIAL COEFFICIENTS AND JACOBI SUMS 433 (12) p = u2 + 6v2, u = -1 (mod 4) if 3\b and u ee 1 (mod4) if 3 a, 24. Although the sign of b is not fixed in (1) it is clear that for/) = 4/ + 1 there exists at least one primitive root g(p) such that ge,/a = a/\b\ (mod p). For e 20 and 24 some of our determinations require fixing g in this manner (see, in connection, [19]). The following congruences are typical of those proved in / =-A (modp = 3f+\) (Jacobi [20]), y ee2û (mod p = 4/4-1) (Gauss [13]), 2/ / U w(x2- I25w2) 2 \ ' 4(xw + 5uv) (mod p = 5/+ 1) (Lehmer[23]), *\ =2x (modp = 6f+ 1), 2x (mod p = 7/+ 1) (Jacobi [19]), (2/My)+($~, <-,=*.). (2/)=(-l)A/42c (mod/, = 8/+l), w(x2- \25w2) -x 4(xw + 5uv) II = -A (mod/>= 12/4-1), (modp= 10/4-1), f)=2a (mod 12/4-1), 4/\l_/_ 3w(x2-13w2) // ~2\X 8(wx+ \3uv) V, '2g(modp=\5f+ 1), (modp= 13/4-1), AB = 0(mod5), a = 1 (mod4) <=» 3 6, 2AA-9B8 (modp= 15/+1),,4 =5 or-25 (mod 5) 2,4g - 18flg,4+95 k{. EE(-l)/2cor(-l)/+12c (mod/> = 16/+ 1) (yh-4.-f'rff \ / / \ mz - vvz + 2«w / (mod /7= 15/4-1), A = -5 or 25 (mod 5), according as b = 0 or 8 (mod 16), ) (N-/-IV+D.

4 434 R. H. HUDSON AND K. S. WILLIAMS 7) + (*') =4(-l)7x (modp=l6f+\), 1), 5\bor5\a (Whiteman [45]), (modp = 20f+ 1) for the solution (x, -v, u, -w) of (4), (5/) = (_!)/+*/«+1 (modp = 24/+l), ee(-1)/2m (mod/? = 24/+ 1). min so far as possible we determine all binomial coefficients for the cases considered if they can be given in terms of the systems (1)-(12). Binomial coefficients which are not treated are related to the parameters in more complicated quadratic partitions. In some cases, see 18-21, we are able to determine these binomial coefficients up to sign in terms of the parameters in (1)-(12). For each ei there are \(e \)(e 2) binomial coefficients of the type (1.1) which are of order e. In 1 it is shown by an application of a simple generalization of Wilson's theorem that it suffices to determine N(e) of these binomial coefficients (for large e, N(e) s e2/12) where N(e) is given explicitly by,,,, i(e2-a)/12 if e = a (mod6), a = -3,0,1,4, (1.3) Afe = \, ) : [(e2 + a)/12 ifeee«(mod6), a = -4,-1. These N(e) binomial coefficients will be called representatives. When e is composite, say e = mn, it is useful to have a congruence of the type i> / o, f\a-,f\ a,f\ (\ 4) n(p-\)/"> = _Li 11-'I (mod p) y ' bj\b2f\---b,f\ Vlw"P> where the a, b are integers between 1 and e 1 inclusive. Such a congruence follows from the Davenport-Hasse relation for Gauss sums [7] and a congruence of Yamamoto [50]; see (3.11). For values of m and n for which m or n is small we show in 4 that the expression on the right-hand side of (3.11) can be given in terms of binomial coefficients (mod p). Together with known results for n(p~^/m (mod p) in terms of representations of p by quadratic forms we deduce congruences (mod p ) relating certain binomial coefficients which are used in later sections. For example, if p = 2mf + 1, it is shown in Theorem 4.1 that we have (,,, 2(,-^E(_1),(<» + 2)/)/(<».2y>/) (modp). Putting (1.5) together with a result of Emma Lehmer [23], see (4.5), we obtain Corollary which is used in 9, 15, and 21. The results in 4 appear to us of

5 BINOMIAL COEFFICIENTS AND JACOBI SUMS 435 some interest, totally apart from their use in later sections. For example, an easy application of (2.1), (2.2) and (3.11) gives *,-.>/«=(_!/(»*/)/ (!3y) (mod p = 48/+ 1). We note that criteria for 3 to be a 16th power in terms of the parameters in Diophantine systems is an open problem. In 1840, Cauchy [5, p. 37] show that o,, (THv) < *=*/+!). and in 1965 Whiteman resolved the sign ambiguity in this congruence. Representative binomial coefficients which are congruent to ± 1 times another representative modulo p for all p ef + 1 are said to be of Cauchy-Whiteman type for reasons given in [17] and 21. We systematically investigate all such congruences. The 27 congruences of this type given in Theorems 21.1 and 21.2 far exceed the number of such congruences for all e < 24. Moreover, the congruences in Theorem 21.2 do not arise (as do all other known Cauchy-Whiteman type congruences) as expressions of the form («<'" 1)/m)',/> = mnf+ 1. For/7 = 11/ + 1, 4/7 = a2 + 1 lb2, a ee 2 (mod 11), Jacobi [19] showed that (M/(îfl-. For larger values of e with many representative binomial coefficients it is not appropriate to list all such congruences, and we cite only one. For/7 = 20/+ 1 =e2 + 5/2 (e = a (mod5) if 5 \b and e =\b (mod5) if 5 a) we have (fl(y)/(*).(-,r^<m,i, The sign may be given unambiguously in this congruence (and in many other congruences in 20) only because of an important sign ambiguity resolution obtained by Muskat and Whiteman [31] in determining the cyclotomic numbers of order 20. In all that follows we are heavily indebted to Berndt, Evans, Muskat, and Whiteman for their pioneering work on Jacobi sums of higher orders. 2. The number of distinct binomial coefficients of the type (1.1). If m and n are positive integers such that m + n = e, then a simple modification of Wilson's theorem yields (2.1) m/!«/! = (-ir/_,=(-l)n/"1 (mod/7).

6 436 R. H. HUDSON AND K. S. WILLIAMS Making use of (2.1) and the elementary property ( ) = (a-h) of binomial coefficients, we deduce that for 1 < s < r < e 1 we have «(i)-((r-^)-<-h{:-ä(-,r*((-ä s<-i»"((v")//)s(-i)"((e":/+s,/)<mo<""- If r 7e 2s, 2r s = e, r + s = e then the entry pairs in the six coefficients in (2.2) are distinct and we call (2.2) a 6-cycle. If exactly one of r = 2s, 2r s = e, r + s = e holds, then there are three distinct entry pairs in the coefficients in (2.2) and we call (2.2) a 3-cycle. Finally, if at least two of r 2s, 2r s = e, r + s = e hold, then in fact all three hold so that e = 3s and the coefficients in (2.2) reduce to the 1 -cycle (2//). The number TV, of 1-cycles is clearly 1 if e = 0 (mod 3) and 0 if e z 0 (mod 3). The number N3 of 3-cycles is the number of pairs (r, s) satisfying exactly one of r = 2s, 2r s = e, r + s = e. Thus 7v"3 is the number of integers t satisfying 1 < t < e/2, t i= e/3, so that e/2-2 ifeeeo (mod6), (e- l)/2 if e = 1 (mod6), e/2-1 if<?ee2 (mod6), (2'3) ^3" I (c - 3)/2 ife = 3 (mod6), e/2-1 ifi>ee4 (mod 6), (e- l)/2 if >EE5 (mod6). The number N6 of 6-cycles is now easily deduced from the values for TV, and N3 and the identity (2.4) TV, + 3/V3 + 6/V6 = {(e - \)(e - 2). Since the coefficients in (2.2) are congruent (mod/7) it suffices, for each e, to determine N(e) = TV, + 7V3 + N6 of them. For the convenience of the reader Table 1 is given summarizing the above and indicating the representative to be chosen from each cycle. A representative binomial coefficient of the type (1.1) with (r, s) > 1 is the same as the lower order binomial coefficient (rs\j ), where rx = r/ (r,s,e), s, = s/ (r, s,e), ex = e/(r, s, e) < e, /, = (r, s,e)f, p = exfx + 1. Henceforth, a representative binomial coefficient will be said to be of order e only if it is not the same as one of lower order. It is easy to see that if 5(e) denotes the number of representatives with lower order ones excluded, we have 5(3) = 5(4) =1, 5(5) = 5(6) = 2, 5(7) = 5(8) = 4, (2.5) 5(9) = 5(10) = 6, 5(11) = 10, 5(12) = 8, 5(13) = 14, 5(14) =12, 5(15) = 5(16) = 16, 5(20) = 24, 5(24) = 33.

7 \ \ % ki to k oc a. kl a; + I + = + I + ^ ^ + Sp + * i!s> «1 CM < 4 1 t \ wi r^ on m m m o> I I 'S cs m in no oo

8 438 R. H. HUDSON AND K. S. WILLIAMS We close this section with a lemma which will be useful when 5(e) > 1 in that it makes it possible to further reduce the number of binomial coefficients which must be treated separately. Lemma 2.1. Ifg, h, k are integers satisfying 1 < A < g < e 1, \<h<k^e 1, e k < g h, then gf\l(e~ g)f\ _, i«g+k)f(kf\ (e-k)f 3. Basic properties of Gauss and Jacobi sums. For a positive integer n we set f = exp(27t/'//i). For (a, e) = 1, we define the automorphism oa by (3.1) Vf.-C, a-p-»pa, where 5 denotes any of the <i>(e) prime ideals dividing p in (?(fe). Let g be a primitive root (mod p) such that g(p~i)/e = ff (mod 5). We define a character xe (mod />) of order e by { } XÁX) [0 ifxee0(mod/7), C iîx^o(modp),x^-^e = ^(modp), so that xe(a>) = V If * z 0 (mod p), the index of x with respect to g, written indg(;c), is the unique integer b such that x = gh (mod /7), 0 < b </7 2. Clearly Xe(*) fèndg(x)- Let r and s denote positive integers. The Gauss sum Ge(r) of order e is defined by p-\ p-\ Ge(r) = s xwp = 2 $rd (xx = 0 The Jacobi sum Je{r, s) of order e is defined by Je(r>s)= p-\ Gauss and Jacobi sums are related by x=\ p-\ 2 Xe(x)x'eO - X) = 2 t**w*w-*\ x = 0 x=2 (3-3) Je(r, s) = Ge(r)Ge(s)/Ge(r + s), provided e does not divide r, s, or r + s. Moreover (see, for example, Muskat [30]), we have (3.4) Je(r, s) = Je(s, r) = (-l)"/,(-* - s, s), (3.5) 7e(r, s)/e(-r,-s) =p (r, s, r + s s 0 (mod e)), (3.6) G,(r)Ge(-r) = (-l)*/» (r Z 0 (mod e)), (3.7) Jm(r, s) =/e(r/î, s«) ûe = mn, and if e is prime and e J r, s, or / + s, then (3.8) Je(r,s) = -\ (mod(l-u2).

9 BINOMIAL COEFFICIENTS AND JACOBI SUMS 439 An important relation involving Gauss sums is provided by the Davenport-Hasse relation [7], namely,.,,, Ge(tn)l\"z\Ge(mj) (3.9) Cd'(,"= ' i, \. i=l,2,...,m-l. Further, it follows from the work of Yamamoto [50] that if IIy= {Ge(/)}'> (c an integer) is a unit of Q(Çep ), then e-i and (3.10) nw^r^ïo/^ (modp). j=\ 7=1 Applying this to (3.9) we obtain ntf\l\"z\(mjf)\ (3.11) (/>-!>'/«= /-lv y (mod/7). n^o((m/ + 0/)! Finally, it follows from Stickelberger [41] that if et r, s or r + s, and (Je(r, s)) denotes the ideal generated by Je(r, s), then (3.12) <^.*))= n p, t=\ U,e)=\ {rr'/e) + {st-'/e}<\ where t'x denotes the inverse of t modulo e, and { } denotes, as is customary, the fractional part of the quantity inside the braces. 4. n(p~1)/m as a product of binomial coefficients (mod p). In this section for values of m and n for which either m or «is small and t = 1 we show that the expression on the right-hand side of (3.11) can be expressed as a quotient of products of binomial coefficients (mod/7). Making use of known results in the literature for nip~])/m (mod p) in terms of representations of p by quadratic forms, we deduce congruences (mod p) relating certain binomial coefficients of the type (1.1). These congruences will be used in later sections. Throughout this section we have p = mnf + 1. Taking n = 2 in (3.11) and appealing to (2.2), we obtain for each t = 1,2,...,m 1 and/7 = 2mf + 1, 2<-»^E=(-ir(^)/(^) (mod/7). Binomial coefficients (mod p = e/+ 1), when e is composite, are often related to one another by powers of n(p~1)r/m where «is a divisor of e (not necessarily prime). For e < 12 we are able to determine these interrelationships by simply taking t = 1 in (3.11) so we may appeal directly to results in this section. Beginning with e = 14 in 17 the number of powers of n(p~x^'/m which need be considered becomes

10 440 R. H. HUDSON AND K. S. WILLIAMS sufficiently large that it is convenient to use the full strength of the congruence (3.11). We postpone this generalization for now and it will be understood throughout this section that t = 1 in applying (3.11). Taking n = 2 in (3.11), and appealing to (2.2), we obtain Theorem 4.1. If p = 2mf + 1 is prime then (4.1) 2<>->/»S(l)'( {m~l)f)/[r ff) (mod,). When m 2 we have, as is well known, (4.2) 2ip-l)/2=(-l)f (mod/7 = 4/+ 1). When m = 3, Theorem 4.1 gives (4.3) (-!)'( 2/)=2<'-'>/3(y) (mod/7 = 6/+l). As p is a prime = 1 (mod 3), there are integers x, y such that (4.4) p = x2 + 3y2, x = \ (mod 3). The determination of 2(p~ l)/3(mod p) in terms of x and v is given by (4.5) 2("-'>/3 = 1 (mod p) ifv = 0(mod3) x + 3v (mod/?) x 3v if v = 1 (mod 3) x 3j (mod/7) x + 3 v ifv' EE 2 (mod3) (Jacobi [19]), (Lehmer[23]). Primes p = 1 (mod 3) are also expressible in the form 4p = A , A = 1 (mod 3), where A and 5 are related to x and v in (4.4) by (4.6) A = -2x, B=±2y if v ee 0 (mod 3), y4=x + 3y, 5=±3L(*-v) if y ee 1 (mod 3), y4 = x - 3v, 5 \(x + y) ifv ee 2 (mod 3). Thus (4.5) may be reformulated in terms of A and 5 as follows: l(mod/7) if,4 =5 = 0(mod2), (4.7) 2('-')/3ee. y (mod/7) if,4 = ee 1 (mod2),/< ee 5 (mod4), A 95 (mod/7) if ^ ee 5 ee 1 (mod2),a = -5 (mod4). A + yß Combining (4.3) and (4.5) we obtain

11 BINOMIAL COEFFICIENTS AND JACOBI SUMS 441 Corollary Ifp = 6/+ 1 = x2 + 3v2, x = 1 (mod 3), is prime, then 3/\_ / (-l/jyjonodp) (-l/f^fyjimod/,) */v = 0(mod3), //>>EEl(mod3), Combining (4.3) and (4.7) we obtain (-1)7^ (2/) (m d p) i/vee2(mod3). Corollary then Ifp = 6/+ 1, with 4p = A , A = 1 (mod3), is prime, (-!)'( y) (mod/7), if A =5 = 0 (mod 2), > (-^J^rff (2/) (mod^)' '/^=5EEl(mod2),>i=5(mod4), (_1)/l^lf ( /] (mod ^ lfa=b=l (mod2)> ^ = "*(mod4)- When m 4, Theorem 4.1 gives (4.8) (yj^-l)'^)/4^) (mod/7 = 8/+l). As p = 1 (mod 8) there are integers a and b such that (4.9) p = a2 + b2, a = l(mod4), 7 = 0(mod4), and by a result of Gauss [13, Vol. 2, p. 89] we have 2O--0/4 =(_!)*/< (mod/7). Hence from (4.9) and Lemma 2.1, we have Corollary If p = Sf + I = a2 + b2 (a = I (mod4), 7 = 0 (mod4)) is prime, then (*)-<-.)-$) (nmd,, When m = 5, Theorem 4.1 gives (4.10) (y) ^i-l/^-)/5!^) (mod/7 = 10/+ 1). As p = 1 (mod 5) there are integers x, u, v, w such that (4.11) _2 I6/7 = x2 + 50m2 + 50i w2, x = 1 (mod 5), xw = v2 4uv u2.

12 442 R. H. HUDSON AND K. S. WILLIAMS It is known (see, for example, [49, p. 544]) that (4.11) specifies x uniquely. If x = 0 (mod2), Lehmer [21] has shown that 2(p~X)/i = 1 (mod p). Moreover, if x = 1 (mod 2), then there is a unique solution (x, u, v, w) of (4.11) satisfying (4.12) meeo (mod2), x + u - v = 0 (mod4), and for this solution we have 2(p~1)/5 = a(x, u, v, w)(mod p) where,,, v w(l25w2 - x2) + 2(xw + 5mu)(25w - x + 20m - 10«) (4.13) a(x, u,v,w) = -zr---- w(l25w2 - x2) + 2(xw + 5mu)(25w - x - 20m + 10») Combining these we have Corollary If p = 10/ + 1 is prime, and (x, u, v, w) is a solution of (4.11) satisfying (4.12) we have 4f\- f Ï 2/1 if x = 0 (mod 2), (-l)fa(x,u,v,w) {. (mod/7) if x = 1 (mod2), w/iere a(x, u, v, w) is given by (4.13). When m = 6, Theorem 4.1 gives (4.14) (5/)s(-l/2f-^(^) (mod/7=12/+l). Since 2(i"l)/2 ee (-l)/(mod /,), we have (4.15) (/H2<'~1>/3)2(2/) (m0d^- Appealing to (4.5), we obtain Corollary Ifp = 12/ + 1 = x2 + 3 v2 (x = 1 (mod 3)) is prime, then 6/1. (mod/7) z/y = 0(mod3), 5/\_ / ^f(26^)(mod/7),/^l(mod3), 7^(2/) (mod^) ^ = 2(mod3). When m = 1, Theorem 4.1 gives (4.16) (y)s(-l/2<'-' 7/7/ 2/ (mod/7= 14/+ 1). The determination of 2(p"1)/7 (mod /7) has been given by Nashier and Rajwade [33]. Since this determination is extremely complicated, we just illustrate it below for the case when 2 is a seventh power (mod p).

13 BINOMIAL COEFFICIENTS AND JACOBI SUMS 443 Corollary Let p 14/ + 1 be a prime. Then there are integers x,,...,x6 such that 72^ = 2x2 + 42(x2 + xj + x2) + 343(x5 + 3x6)2, (4.17) \2x\ - 12x2 + \41x\ - 441x x,x6 + 24x2x3 24x2x4 + 48x3.X4 + 98x5x6 = 0, 12x32-12x2 + 49x52-147x x,x5 + 28x,x6 +48x2x3 + 24x2x4 + 24x3x x5x6 = 0, with x, EE 1 (mod7). All the solutions of (4.17), except the two trivial solutions (x,, x2, x3, x4, x5, x6) = (-6/, ±2m, ±2m, =í=2m,0,0), where p t2 + 7m2, t = 1 (mod7), have the same value of x,. // x, ee 0 (mod 2), then 2 is a seventh power (mod p), and we have (4.18) (y)-(-l/ V) (mod/7). Example. We illustrate Corollary by taking p 673 so that /= 48 and x, = 22 = 0 (mod 2) (see [47, p. 1136]). In agreement with (4.16), we have f = (24888)ee346 (mod 673), When m = 8, Theorem 4.1 gives (-1)/(27/) = (3966)"346 <m d673)- (4.19) 7/ /(-l)/2('-,,/'(^) (mod/,= 16/+l). From Lehmer [23, p. 66] we have (4.20) 2<i--')/8 = J + 1 ifz7 = 0(modl6), + b/a if b = 4 (mod 16), -1 if b = 8 (mod 16), -b/a \ib= 12 (mod 16), where a and b are defined as in (4.9). Combining (4.19) and (4.20) we obtain Corollary prime. Then Let p = 16/+ 1 = a2 + b2 (a = 1 (mod4), 6 = 0 (mod4)) >e -(- /(ira (-'> 7/\_ / Since expressions for 2</,_1)/m (mod /?) are also known for m = 10, 12, 15, 16, 20, 24, 32 and 40 (see [23, p. 70; 18]), similar congruences to those given in Corollaries can be deduced.

14 444 R. H. HUDSON AND K. S. WILLIAMS Next we take n = 3 in (3.11) to obtain Theorem 4.2. Ifp = 3m/+ 1 is prime then When m 3, Theorem 4.2 gives (4.21) (y)ee3<"-"/3(2^) (mod/7 = 9/+l). By a result of Lehmer [23, p. 67] (see also [48, p. 279]) we have l(mod/7) if5 = 0(mod3), (4.22) 3<*-0/3 A 95 (mod/7), if5ee l (mod3), A ^ + 95 ir^(mod/7),4-95 if5 = 2(mod3), where (4.23) 4/7 = A , ^ = 1 (mod 3). Combining (4.21) and (4.22) we obtain Corollary Ifp = 9/+ 1 is prime then ( 2/) - ( 5f) (mod ^ ifb = (mod 3)' = ^H(5/)(mod/7),/5ee l(mod3), ^7rlf(5/)(mod/,) '/5 = 2(mod3)- When w = 4, Theorem 4.2 gives (4.24) (6/)-(-3)('",)/4(y) (mod/,= 12/+l). From the work of Gösset [15] we have (p-\)/a _ [ 1 (mod p) -l(mod/7) if ) = 0 (mod3), ifa = 0(mod3), where a = 1 (mod4), 7 = 0 (mod2) ((-3)^"1)/4 = ± 1 (mod p) as/7 = 1 (mod 12)). Combining (4.24) and (4.25) we have Corollary If p = I2f+ \ = a2 + b2 (a = I (mod4), 7 = 0 (mod2)) is /7n'me then V\ - / f^j (mod p) i/ 7 = 0(mod3), M (mod p) if a = 0 (mod 3).

15 When m = 5, Theorem 4.2 gives BINOMIAL COEFFICIENTS AND JACOBI SUMS 445 (4-26) (/)S3('"I)/5(3/) (mod^15/+1)- An explicit determination of 3(p~1)/5 has been given in [49, Theorem 2]. Using this together with Jacobi sums of order 15 given by Muskat [30] an explicit determination of (J) and (Yf) is obtained in 18. Taking n = 4 in (3.11), and appealing to (2.2), we obtain Theorem 4.3. Ifp = 4m/+ 1 is prime then 4(,- /,-/2/\/3/W4/ /xxvirrni y) Taking m = 3 in Theorem 4.3 we obtain Corollary Ifp = 12/+ 1 is prime then (6/)=2<-'>/3(y) {moápy We note that it is possible to obtain a simpler form for the determination of 4<p i)/m (m0(j pj by using (2.1) together with (3.11). In particular, we have ^(p-\)/m = 4/!(-l)m/ '(2m/)! uu ^,wm /!((«+l)/)!((2«+l)/)!((3«+l)/)! II {(mj+ 1)/)! 7 = 0 /(m + 3)/W(m + 2)/ (-l)/4/!(2m/)!(m-l)/! _, / \ f }\ f :(-!)' /!((m+l)/)!((2m+l)/)! / (m + 3)/W (2m + 1)/ Thus we have obtained the following variation of Theorem 4.3. Theorem 4.4. Ifp = 4m/+ 1 is prime then 4P-Wm^{_x)fUrn + 3)f\i(m + 2)f\^ I (m + 3)/W(2m+l)/\ (mod p). Although Theorems 4.1 and 4.4 clearly give the same result for m = 3, this is not the case in general. The following congruence, which will be referred to again in 14, is particularly interesting, since it shows that representative binomial coefficients may be identical modulo p = e/+ 1. Several of these identities are established in 21. It would be interesting to know for which values of e such congruences are possible.

16 446 R. H. HUDSON AND K. S. WILLIAMS Corollary Ifp 12/+ 1 is prime then (5/h:í) (mod p ). Proof. Using (3.4) together with Theorems 4.1 or 4.4 (appealing to (2.2) to see that (-\)f(6/) = C/) (î)/(ï)-(ï)/(r (mod p) in the latter case), /(SMS)/(?) <-'> On the other hand, (mod p ) is an immediate consequence of Lemma 2.1 with g = 4, h = 2, k 6. We remark that one can also obtain the last step in the proof by expanding in terms of factorials using (2.1). This corollary is important to us, as using it, other representative binomial coefficients of order 12 are determined in 14 using the simple determination of (2/),p = 3/ + 1, given by, e.g., Jacobi [20]. Taking n = 5 in (3.11), and appealing to (2.2), we obtain Theorem 4.5. Ifp = 5m/+ 1 is prime then 5(/>-l)/m = (m + 2)/\ / (3m - l)/\ / / (m + 4)/\ / (3m + 1)/) 5/ (m + 4)/ / f\(m + 2)f}' \ Taking m = 3 in Theorem 4.4 we obtain (4.27) 5-M = 5< s^-' \J^ (mod/7=15/+l) / By a theorem of Williams [48, pp ] we have (mod p ). (4-28) 5</>-n/3 1 (mod p ) if^5ee0(mod5), A + 95 Q (mod/7) 95 if A = B or -25 (mod 5), 95 (mod/7) A + 95 if A = -5 or 25 (mod 5), where 4/7 = A , A = 1 (mod 3). Combining (4.27) and (4.28) we obtain Corollary Ifp = 15/+ 1 is prime then If" (mod/7), if AB = 0 (mod 5), 2/ ^H ^4 + 95, 72f\ (mod p), if A =Bor -25 (mod 5), A-9B (If A + 9B\2f (mod p), if A EE -5 or 25 (mod 5).

17 BINOMIAL COEFFICIENTS AND JACOBI SUMS 447 Finally, taking m = 4 in Theorem 4.4, we obtain Since (7) 5<"~1)/4( 3 /) (mod/, = 2 /+l). 5(í-D/4 = f1 (mod/7), if >EE0(mod5), -l(mod/7), if a ee 0 (mod5), where a = 1 (mod 4), we have Corollary Ifp = 20/ + 1 is prime then io/\_ / ^ (mod/?) 3/ (mod/7) ifb = 0(mod5), if a = 0(mod5). Corollary was first proved by Whiteman [45], although the congruence (l0/) ee ±(3 /) (mod p) had been established by Cauchy [5,p. 37] one hundred and twenty-five years earlier. 5. The basic theorem. We prove the following theorem which shows how each binomial coefficient of type (1.1) can be determined modulo P by means of Jacobi sums. This theorem provides a basic tool which will be used through the rest of the paper. Theorem 5.1. If p = ef + 1 is prime and r, s are integers such that 1 < s < r < e 1, then (5.1) (^)=(-ir/+ve(r,e-s) (mod5). Proof. Since we have However, (5.2) Xe(x)=x' (mod 5) p-\ Je(r, e - s) = 2 *r/0 - *T~S)i (mod 5) x=\ p-\ (e-s)f 1-1 ye-s)j /,. v 2*r/ 2 (-1)' (e_i)/x' :=I f = 0 \ Í / 2Vi)'((e~5)/) =n \ / / r=i 'ïx*+: _[0(mod/7) if Â: s 0 (mod p - 1), x=i 1 (mod p) if A: = 0 (mod p 1),

18 448 R H HUDSON AND K. S. WILLIAMS so that we have, appealing to (2.2), (5.3) Je(r,e-s)JeÍ (-1) i = 0 l(e-s)f) I (e - s)f\ _,,vf+\[rf We note that Whiteman [45] has already proved a result similar to, but not exactly the same as Theorem 5.1. Letting ß = e2v,/e be replaced by g for a primitive root g of p = e/+ 1 our Je(r, s) becomes Whiteman's \pr s. In Lemma 6 of [45] Whiteman showed that (i) 4>rs = 0(modp) (r + s<e), I (2e - r - s)f\ (ii) «rv,*-- (e-r)f j ^mod p^ (r + s>e). In view of (2.2), condition (ii) can be rewritten in the simpler form,s/+l (ii)' *(r.í)=(-l) """ Lj^J (mod/7). In later sections we will refer again to Whiteman's very useful Lemma e = 3. There is a single representative binomial coefficient of order 3, namely (2f). With A and 5 defined as in (4.23), we choose P (it), where (6.1) 'ÏÏ = \{A +35/^3), so that P1/7. It is well known that (6.2) y3(l,l) = 77, (see, for example, [4, p. 357]), so by (6.1) and (6.2) (6.3) /3(2,2) = v = ^(a - 35/^3) ee^i (mod 77). rf Hence, by Theorem 5.1 (with e = 3, r = 2, s = 1) we have (as/is even), (6.4) y ee -y3(2,2) ee -A (mod 77). As (2J) and -A are both rational integers, and ir\p, we have Theorem 6.1. If p = 3/+ 1 is prime and A is given uniquely by 4p = A , /I ee 1 (mod 3), then y) =- (mod/7). This result is due to Jacobi [20]; see also Whiteman [42] and von Schrutka [35]. Thus, appealing to (4.6), Theorem 6.1 can also be given in the form

19 BINOMIAL COEFFICIENTS AND JACOBI SUMS 449 Theorem 6.2. If p = 3/+ 1 is prime and x is given uniquely by p = x2 + 3v2, x ee 1 (mod 3), then 2/ / 2x (mod p), if y = 0 (mod 3), -x 3v (mod p), if y = 1 (mod 3), -x + 3v (mod p), if y = 2 (mod 3). 7. e = 4. There is a single representative binomial coefficient of order 4, namely / (2/). / With a = 1 (mod 4), b = 0 (mod 2), we choose 5 = (77), where 77 = a + 7/', so that P1/7. Then it is known that./4(l,2) = (-l)/+l77 (see, for example, [4, p. 361]), so /+>=-/ n/+'/- _ «..-ï =/ n/+'- 4(2,3) = /4(1,2) = (-1)' v = (-ir'(a - bi) =(-\)^l2a (mod 77). Thus, by Theorem 5.1, we have 2/) EE(-l)/+ly4(2,3)EE2a (mod 77), and hence Theorem 7.1. If p = 4/+ 1 is prime and a is given uniquely by p = a2 + b2, a = \ (mod 4), then - 2a (mod p). This is the result of Gauss mentioned in 1; see also Whiteman [42, p. 95]. 8. e = 5. There are two representative binomial coefficients of order 5, namely ( ) and (V). For convenience we set ß = f5. It is known that the ring of integers 5 or Q(ß) is a unique factorization domain [27]. In 5, p factors into primes as (8.1) p 77, , where 77 is any prime factor of p in 5 and mi = 0,(77) (i = 1,2,3,4). We can set (8.2) n = axß + a2ß2 + a3ß3 + a4ß4, where a a2, a3, a4 are rational integers (see, for example, [49]). Clearly a, + a2 + a3 + a4 2 0 (mod 5), as 1 ß \ 5, 5 \p. Replacing 77 by its associate 077, where a is the unit of 5 given by (8.3) + 1 if a, + a2 + a3 + a4 = 1 (mod 5), -(ß + ß4) if a, + a2 + a3 + a4 = 2(mod5), + (ß + ß4) if a, + a2 + a3 + a4 ee 3 (mod5), -1 if a, + a2 + a3 + a4 ee 4 (mod 5), we may suppose that 77 = 1 (mod(l ß)). Replacing the new value of 77 by its associate ß-<"i+2a2+3a3+4a4)77i we may SUpp0se further that (8.4) 77 = 1 (mod(l-ß)2).

20 450 R. H. HUDSON AND K S. WILLIAMS By a theorem of Stickelberger, see (3.12), we have (8.5) /S(1,1)=0 (mod77,773), so (8.6) 4(1,1) = «77,773, where m G 5. From (3.5), (8.1) and (8.6) we have SO MM = ( M77,773 )( M77,773 ) = J ( 1, 1 ) J$ ( 1, 1 ) =/7 = 77, , (8.7) MM=1, showing that m is a unit of 5, that is (see, for example, [36]), (8.8) M=±(ß + ß4)V (A: = 0,±l,±2,...;/ = 0,1,2,3,4). Now (8.7) guarantees that k = 0 in (8.8) so (8.9) u=±ß' (/ = 0,1,2,3,4). By (3.8), (8.4), (8.6) and (8.9), we have ±ß'=u = M77,773 (mod(l ß)2) = 4(1,1) (mod(l-ß)2) EE-1 (mod(l-ß)2), so in (8.9) the minus sign holds with / = 0, that is, u = -1, giving (8.10) 4(1,1) = -77,773. We set (8.11) 4(1. 0 = c,ß + c2ß2 + c3ß3 + c4ß4. As 4(1,1) ee -l(mod(l -ß)2), by (3.12), we have Je, +c2 + c3 + c4ee -1 (mod5), * " ' \cl + 2c2 + 3c3 + 4c4ee0 (mod5). Next, since ß - ß2 - ß3 + ß4 = /5, we have (c, -c2-c3 + c4)^5=(ci -c2-c3 + c4)(ß - ß2 - ß3 + ß4) -(1+c,+ c2 + c3 + c4) (mod(l-ß)4) ee 2((c, + c4)(ß + ß4) + (c2 + c3)(ß2 + ß3) + 2) (mod(l - ß)4) ee 2(4(1,1)+ 4(1,1) +2) (mod(l-ß)4) = 2(4(1,1)+ 1)(/5(1,1) + 1) (mod(l-ß)4) eeo (mod(l-ß)4),

21 BINOMIAL COEFFICIENTS AND JACOBI SUMS 451 SO (8.13) c, - c2 - c3 + c4eeo (mod5). Congruences (8.12) and (8.13) enable us to define integers x, u, v, w by (8.14) x = - (c, + c2 + c3 + c4), 5m = c, + 2c2 2c3 c4, 5 c c2 + c3 2c4, 5w = c, c2 Using (3.5), (8.11), (8.12) and (8.14), it is easy to check that (x, u, v, w) is a solution of (8.15) 16/7 = x2 + 50«2 + 50u w2, x = 1 (mod 5), xw = v2 4uv u2. From (8.14), we obtain (8.16) 4c, = -x + 2m + 4v + 5w, 4c2 -x + 4u 2v 5w, and so (8.11) and (8.16) give c3 -x 4m + 2v 5w, 4c4 = -x 2m 4v + 5w, (8.17) 4(1,1) = ^(x + M(2ß + 4ß2-4ß3-2ß4) Next, from (8.17), we deduce that + u(4ß-2ß2 + 2ß3-4ß4) + 5w{5). (8.18) 4(1,1)+4(4,4) = ^(x + 5w/5). Since (8.19) 4(1,1) =0 (mod 77), by (8.5), we deduce from (3.4), (8.18) and (8.19) that (8.20) 4(2,4) = 4(4,4) =\(x + Sw{E) (mod 77). Hence, by Theorem 5.1, we have (8.21) y ) =-4(2,4) = -!(* +5W5") (modtr). It now remains to determine ^5" (mod 77) in terms of x, u, v, w. Since (8.22) ß + 2ß2-2ß3 - ß4 = \ij50 + IO/5, 2ß - ß2 + ß3-2ß4 = -iy50 - IO/5, we obtain from (8.17) and (8.19): (8.23) x + imi/50 + IQ/5 + ivjso - IO/5 + 5w/5 ee 0 (mod 77)

22 452 R. H. HUDSON AND K. S. WILLIAMS Also from (8.5) we have (8.24) 4(1,1)=0 (mod773). Applying the automorphism a2 to (8.24), we obtain (8.25) 4(2,2) eeo (mod 77,). Hence from (8.17) and (8.22) we have (8.26) x - iu] 50 - IO/5 + /o^50 + IO/5-5w/5 ee 0 (mod 77). Adding (8.23) and (8.26) we obtain (8.27) 2x + /(«+ o)^50 + IO1/5 - i(u - v)]j50-10/5 ee 0 (mod 77). Taking the term 2x over to the right-hand side of (8.27) and squaring, we obtain after some simplification, (8.28) 10(m2- uv- v2)fi = x2 + 25m2 + 25o2 (mod 77). From (8.15) and (8.28), we obtain (8.29) /5 ee -(x2 + 25M2 + 25tr)/10(xw + 5mo) (mod 77). Using (8.29) in (8.21), we get, appealing to (8.15), (y)-l+y-'f) \ f I L ö(xw + 5mü) h,). As both sides of the congruence (8.30) are integers (mod p), and since x, x2 125w2 and x + 5uv/w are independent of the choice of solution (x, u, v,w) of (8.15), (8.30) holds mod p. Similarly, using J5(2,2) in place of J5(l, 1), we obtain an analogous congruence to (8.30) for (3^). These congruences are due to Emma Lehmer [23, p. 69]. Summarizing, we have Theorem 8.1. Ifp = 5/+ 1 is prime and (x, u, v,w) is any solution 0/(8.15), then (Y)5lL+"-y-'25-;>) \ / / 2 \ 4(xw + 5ho) / (y-il.-'y- *?> (mod,). (mod,». \ // 2\ 4(xw + 5mo) / V ^7 The next corollary follows immediately rediscovered by Rajwade [34]. Corollary from Theorem 8.1. It was recently Ifp = 5/ + 1 is prime and x is given uniquely by (8.15), then x+(y) + (y)=o (mod/,).

23 BINOMIAL COEFFICIENTS AND JACOBI SUMS e = 6. There are two representative binomial coefficients of order 6, namely (2f) and (3/). In this section we establish a congruence for (2f) which, in conjunction with Corollary 4.1.1, gives (9.1) I y = 2x (mod/7 = 6/+ 1). We have been unable to find a reference to this result. Consider the Jacobi sum 4(2,5). By (3.4) and a result of Jacobi [19, p. 69], we have that is (by (3.7)) 4(2,5) = (-l)/4(5,5) = x '(4)4(4,4), 4(2,5) = X3'(2)4(2,2). Since x3(2) ee 2(p~ l)/3 (mod 77) from (4.7) and (6.3) we obtain 'A (mod 77), if,4 = 5 = 0(mod2), 4(2,5) -]r(a + 95) (mod 77), if A =5 = 1 (mod2)m = 5(mod4), -Ua -95) (mod 77), iía=b=\ (mod2),/4 = -5(mod4). Thus by Theorem 5.1 we have Theorem 9.1. Ifp = 6/ + 1 is prime and A, B are defined by (4.23), then 2/ / '(-\)f+la(modp) if A ee 5 = 0 (mod 2), (]_ f-(a + 95) (mod p) 2 J_ if A = B = 1 (mod2),,4 =5(mod4), (-\YUa -9B)(modp) ifa=b = \ (mod2),/l ee -5(mod4). 2 Appealing to (4.6), we obtain Theorem 9.2. Ifp = 6/ + 1 is prime and x, v are defined by (4.4), then 2A - / 2(-\)fx(modp) />EE0(mod3), (-\)f(-x + 3v) (mod p) ify = \ (mod3), (-l)f(-x-3y)(modp) if y = 2 (mod 3). Example. We illustrate Theorems 9.1 and 9.2 by taking p = 991, so that/= 165, x = 22,v= 13,,4 =61,5 = 3. We have ~(\f5) =914= -17 (mod 991), (_l/(_x + 3v) = (-l)f\(a -95) = -17.

24 454 R. H. HUDSON AND K. S. WILLIAMS 10. e 1. There are four representative binomial coefficients of order 7, namely, By (2.2) and Lemma 2.1, we have so that it suffices to determine m - (r modulo/7. In order to do this by means of Theorem 5.1 one must consider the Jacobi sums 4(2,6), 4(3,6) and 4(4,6), respectively. Of these, 4(3,6) is an integer of the subfield Qfñ of g(f7), as a2(4(3,6)) =4(6,5) =4(3,6), and we are able to reprove Jacobi's result [19] for (3f) (mod p) using Theorem 5.1. The other two Jacobi sums are related to 4(1,1) by 4(2,6) = 4(6,6) = a6(4(l, 1)), 4(4,6) = 4(4,4) = o4(j7(\, 1)), so that to determine (2/) and (*/) modulo p it suffices to consider 4(1,1). This Jacobi sum, unlike 4(3,6), does not belong to a subfield of ö(f7). We are able to express 4(1,1) in the form C,f7 + C2f72 + C3f73 + C& + G 75 + Qf76 where the C, i = 1,...,6, are linear combinations of a non trivial solution (x,,...,x6) of (4.17). Using Theorem 5.1 we are able to obtain the congruence where If ) = -2(2x, + 7x55 + 2lx6S) (mod 77) 5 = f7 + tf - 2f3-2tf + f75 + tf, S = f7 - J72 - f75 + f76, and 77 denotes any prime factor of p in the ring of integers of ô(f7), but, unfortunately, we have not been able to determine 5 and S mod 77 in any aesthetic form. Consequently, unlike the case e = 5, we are unable to give (2/) and (jf) mod p explicitly in terms of invariants of the system (4.17), although a result analogous to Theorem 8.1 (but more complicated) may well exist. We are (in analogy to Rajwade's result [34]) able to evaluate 2 H H>^- First we show, however, how Theorem 5.1 can be used to deduce Jacobi's result [19]. The ring 5 of integers of ß(f7) is a unique factorization domain [27]. In 5, p factors into primes as (10.2) /7 = 77, ,

25 BINOMIAL COEFFICIENTS AND JACOBI SUMS 455 where 77 is any prime factor of p in 5 and 77, = 0,(77), i = 1,2,3,4,5,6. analogy to the case e = 5 (see 8.4) we may normalize 77 so that In precise (10.3) 77EEI (mod(l-f7)2). By (3.12) we have St) 4(1,2) =0 (mod 77,772774) 4(1,2) = wnxm2m4, where u is an integer of 0(f7). In view of (3.5) we have mm = 1 so u is a unit of 0(f7). As all units of 0(f7) are of the form (10.4) ±(ß + ß6)k'(ß2 + ß5)klß' (see, for example, [36, p. 99]), it follows from (10.4) that, = k2 = 0, therefore (10.5) u=±ß', = 0,1,2,3,4,5,6. But 4(1,2) = -1 (mod(l - f7)2) and 77, ee 1 (mod(l - f7)2) so MEE-l (mod(l -f7)2). Thus (10.5) must hold with the minus sign and with 1 = 0, that is, Next, as 4(1,2) = -77, (4(1,2)) = a2(-77,772774) = r, =4(1,2), we deduce that 4(1,2) 0(V-7~). Since 4(1,2) is an integer of 0(f7), it must be an integer of Q(4-l), so there are integers X and Y with X = Y (mod 2) such that (10.6) J1(l,2)=^(x+Ypî). As 4(1,2)4(1,2) = p, by (3.5), we have (10.7) 4/7 = X2 + 7y2, which implies there exist integers x and v with X = 2x, Y = 2 v, and (from (10.6) and (10.7)) 4(1,2) = x +yfï, x2 + ly2=p. As 4(1,2) = -1 (mod 1 f7)2 and (as is easily checked), fl = Í, + f72 - tf + tf - f75 -?7 = 0 (mod(l - f7)2), we have x = -1 (mod(l f7)2) so x = -1 (mod 7). Finally, using Theorem 5.1 we have (10.8) ( y) = -4(3,6) ee -4(5,6) ee -4(172) ee - (x v/-7 j ee-2x (mod 77).

26 456 R.H.HUDSON AND K.S.WILLIAMS As the quantitites in the congruence (10.8) are rational integers, we have the following theorem due to Jacobi [19]. Theorem Ifp = 7/ + 1 is prime and x and y are integers with p = x2 + ly2, x = -1 (mod 7), then - -2x (mod p). We now show that a result analogous to that of Rajwade [34] follows easily from Theorem 5.1 and the basic properties of Jacobi sums listed in 3. First, note that a precisely analogous argument to the one above for 4(1,2) gives 4(1,1) =-77, so 4(1, l) = 4(2,2) = 4(3,3) = 0 (mod77). By Theorem 5.1 we have (10.9) if ee -4(2,6) ee -4(6,6) (mod77), (10.10) jyj ee-4(4,6) ee-4(4,4) (mod77), (10.11) l42ff =-4(4,5) ee -4(5,5) (mod77). Adding ( 10.9)-( ) we obtain (y)+(y)+(^)=-^7(m) (mod,7). Since 6 2 4(', 0 ~ *i» x, ee 1 (mod7), we have Theorem If p = 7/ + 1 is prime and (x,,...,x6) is a solution of (4.17) with x, ee 1 (mod 7), then IH'MZ +, +U, ="*i (mod/7). Example. We illustrate Theorem 10.2 by taking/? = 29 so that/= 4 and x, = 1 (see [47]). In agreement with Theorem 10.2 we have, for/ = 4, (2f)+r/)+(2/) -i ^-] (m d29)- 11. e = 8. There are four representative binomial coefficients of order 8, namely, 2/)'(T)-(4/) - \l

27 Now, by Corollary 4.1.2, we have BINOMIAL COEFFICIENTS AND JACOBI SUMS 457 (11.1) (/H'^lv) (m d^ and appealing to Theorem 7.1, we obtain Theorem If p = 8/+ 1 is prime and a is given uniquely by p = a2 + b2, a ee 1 (mod 4), then (3/)-(-l)/+ft/42«(mod/7). Next, from Lemma 2.1 and (2.2), we obtain Thus from (11.1) and ( 11.2) we have (11.3) (vh-^i4/) (m d^- Again, appealing to Lemma 2.1 and (2.2) we have which gives, in view of (11.3), (11.4) (2/)"(-1)/W4(4/) {m0dp)- (11.3) and (11.4) show that it suffices to determine (4/) (mod p). In order to do this we must consider 4( 1,4). We set ß = fg = (1 + i)/ \ 2. The ring 5 of integers of 0(ß) is a unique factorization domain [27], Let 77 denote a prime factor of p in 5. We have «ih\ a3(4(l,4))-4(3,4)- 4)i -IC\*\- G*^G*W - Gb(1)^(4) -4(1,4), _,. so 4(1,4) belongs in the subfield 0^-2 of 0(ß). As 4(1,4) is an integer of 0( ß), it must be an integer of Q({^2). Thus we can set (11.5) 40,4) = -(c + dfï), where c and d are integers. As 4(1,4)4(1,4) we have (11.6) I -) - 1 ee 0(mod2) ûp\\-n. = p, we have p = c2 + 2d2. Clearly, Further, since 4-2 = ß(i + i), w^ have (11.7) x8(«)-l=0(modf2) if(-) = +l,

28 458 (11.8) R. H. HUDSON AND K. S. WILLIAMS Xtt(n)-ß = 0 (modfl) if (^) = -1. We now combine (11.6)-(11.8) and note that p-\ 2 n=7 1 -n 1, (ï)-- + 1, 2 i n = 2 2(í-3), ( +1, It clearly follows that (11.9) 4(1,4) =-1 (mod2/^2), so (11.10) Then by Theorem 5.1, we have c = 1 (mod 4). (4/)^(-l)/+'^(4T7y ee(_1)'+14(l4) (mod*) (mod77) ee(-1)/(c-í /^2) (mod 77). But 4(1,4) EE 0 (mod 77) by (3.12), so c + df^2 = 0 (mod 77). Hence (4/) ee^o^c (mod 77). Thus we have proved Theorem If p = 8/ + 1 is prime with a and c defined uniquely by p a2 + >2 = c2 + 2d2, a = c = l(mod4), (y)=(-r/2c (2y)-(-D/+i/42c (mod/7), (mod/7), (y)=(-l)"/42c (mod/7). The first congruence in Theorem 11.2 is due to Jacobi [20] and Stern [40].

29 BINOMIAL COEFFICIENTS AND JACOBI SUMS e = 9. There are six representative binomial coefficients of order 9, and using (4.21) it is easy to show that all six are expressible in terms of 7M3;m5;)- In particular, (12.1) (2/)S3('~1)/3(4/) (m0d/,)' (12.2) jyj=3(,-.)/3j4/j {modpl (12.3) (/H30,~1)/3(2/) {moúp)- Unfortunately, we have been unable to determine any of these binomial coefficients explicitly. However, we are able to prove the following theorem analogous to a result of Jacobi; see Theorem Theorem Forp = 9/ + 1, 4/7 = A , A = 1 (mod 3), we have (2/)(T) Proof. Since /(24//HM/(3/H 4/1/5/1 / /3/\ _ 4/!5/!6/! f)\2f}< /(3/)\fj 3/!3/!6/!' the result follows immediately from (2.1) and (12.1) (12.3). <-» 13. e = 10. There are six representatives binomial coefficients of order 10, namely, 2/\ /3/\ I4f\ I5f\ I5f\ (6f\ (13-1) \fv\fv\fv\fv\vv\yv We show that all of the binomial coefficients in (13.1) can be determined from the lower order binomial coefficients (2^) and (62ff) which are given explicitly in Theorem 8.1. We begin by taking the Davenport-Hasse relation (3.9) with e = 10, m = 5, t = 3, to obtain By (3.3) we have G.o(5)G,o(6) = x35(2)g10(3)g,0(8). 40(5,8) = G,0(5)G10(8)/G,0(3) = X35(2)G10(8))2/G10(6) so, by Theorem 5.1, we have = X35(2)4o(8,8) = x35(2)4o(8,4) 5/\- 2/ = (2(p-.)/5)3 4/j (modw))

30 460 R. H. HUDSON AND K. S. WILLIAMS where 77 is defined as in 8, so 5/ (>i(p-1)/5\3 4/ (13.2) \2f)={1(P~W5)\2f) {m0dp)- From (4.10) and (13.2) we have (13.3) (4/)s(-i/(2('",)/s)4(i/) ( dpy Applying Lemma 2.1 (with g = 5, h = 2, k = 4) we have <»*> (SKOVß) 0-,). Using (13.2) in (13.4) we obtain (13.5) (y)3^1^"1^!^) (mod/,)' Applying Lemma 2.1 (with g = 3, h = 1, k = 4), using (2.2), (13.3), and (13.5), we obtain (13.6) (3/H2<,,~,)/5)3(26/) {m dp)- Next, applying Lemma 2.1 (with g = 3, h = 1, A: = 5), using (2.2), (13.4) (13.6), we get (13.7) (y)=2(,-0/5j6/j {modp) Finally, applying Lemma 2.1 (with g = 2, h = 1, k = 5) and using (13.7) we obtain (13.8) (yj^-l)'^*-1'/5)2^) (mod/')- Combining (13.2), (13.3), and (13.5) (13.8), we have the following new theorem. Theorem Let p = 10/+ 1 >e a prime and let (x, u, v, w) be a solution of (8.15). If 2 is a quintic residue of p (equivalently, x is even), we have 2/U(_1)/(3/)s(_I)/(V)s(1)//6/ ' \fl v x w(x2 <x2- - \25w2) 125w2 1 H); ~T -,/... / (mod/7), 8(xw + 5mü) // ' \2f) jc x w(x2 h>(x2 - \25w2 125w2) 1 = (-0 ~T+ 0/-..-L.C,\ (mod/7). 2 8(xw + 5 mu)

31 BINOMIAL COEFFICIENTS AND JACOBI SUMS 461 // 2 is a quintic nonresidue of p (equivalently x is odd), we can choose a solution (x, u, v, w) of (8.15) satisfying u = 0 (mod2), x + u v = 0 (mod4) so that (see Lehmer [21]) 2{p~X)/s=a(x,u,v,w) where a = a(x, u, v, w) is given by (4.13). 77ie«we have (mod/?) -1 t\f i( -x w(x2 - \25w2) \.. = H «h-7t,-,, x (mod/7, \ 2 8(->ctv + 5wt)) / -(- Mf+y"+'fi)) \ ^ 8(xw + 5md) / (mod")- We close this section with two examples illustrating Theorem Example. Let p = 151 so that/ = 15 and 2 is a quintic residue of p. Then (5)-(S)-(5)-(3)-» <-d'5"' (S)-(S)-(SMS)-* ^151>- and A solution (x, u, v, w) of (8.15) with x even is given by (x, u, v, w) = (-4,2,2,4). In agreement with Theorem 13.1, we have,,.// x w(x2-\25w2)\ -4 4( ), J1C1. (-l)[-2- Z(xw + 5uv)) = T+ 8( ) ~52 (m dl51)' // x w(x2- \25w2)\ _ -4 4( ) _ (-1M"2+ 8(xw + 5M,) J-T" 8( ) = 95 (m dl51)- Example. Let /7 = 11 so that / = 1 and 2 is a quintic nonresidue of p. Note that a = 4 so a2 EE 5 (mod 11), a3 = 9 (mod 11), a4 ee 3 (mod 11). Now it is easily checked that and, similarly, (îmîm.mî)-* <>«*»> (im!mm)-< *-»> Moreover, solutions of (8.15) are (x,u,v,w) = (1,0,1,1), (1,-1,0,-1), (1,1,0,-1), (1,0,-1,1).

32 462 R. H. HUDSON AND K. S. WILLIAMS The first of these solutions satisfies (4.12) (u = 0 (mod2), x + u v 0 (mod4)) and, in agreement with Theorem 13.1, we have and,,,/-,/ x w(x2-\25w2)\ ( \, JllX (-1)V(-2 - (xw + 5uv) r-5h - 8(ÏTO)) "2 (m0du) (-l)vu - y-1^) - -3Í-1 + ±=J* ) EE4 (modll). v ; \ 2 &(xw + 5uv) ] \ 2 8(1 + 0) / v ; 14. e = 11. There are ten representative binomial coefficients of order 11, namely 2/\ /3/\ /4/\ /5/\ /6/\ i/n/n/n/n/ 4/\ /5/\ (6/\ /6/\ llf\ 2/)'U//'\2//'l3/)'U//' It appears to be difficult to determine any of these explicitly modulo p in terms of the variables of a quadratic partition of p such as (14.2) 4/7 = a >2, a = 2 (mod 11), or the representation given in [25]. We first show that Theorem 5.1 can be used to reprove a theorem of Jacobi [19] relating (3/), (\ff) and (3^) modulo p. Let 77 be a prime factor of p in the unique factorization domain 5 of integers of ô(fn). By Stickelberger's theorem (3.12), we have 4,(1,2) ~ 77, g, 4l(2,2) ~ 77, g, 4l(3,3) ~ 77,7r , where, if a, and <x2 are integers of ß(fn), a, ~ a2 means that a,/a2 is a unit of the ring of integers of ô(f,,). Hence, (14.3) Y =4,(1,2)4,(3, 3)/4,(2,2) ~ 77, , showing that y is an integer of ô(fm). Next, appealing to (3.3), we have so Y = G (1)G (3)G (4)/G (2)G1,(6), o3(y) = G (3)G (9)G (1)/G,,(6)G (7). Since, by (3.6), G,,(2)G,1(9) = G,,(4)G,,(7) =p, we obtain a3(y) = y, which shows that y belongs in the subfield Ô(/-ÏT) of ô(f,,). As y is an integer of Q(ÇU), it must be an integer of ß(V-ll ), and so has the form (14.4) Y = -^(û + ^/IÎT), where a, 7 are integers such that a = b (mod 2). From (14.3) and (3.5) we have YY = P- Hence a and 7 satisfy the equation given in (14.2). The congruence in (14.2) follows as by (3.8). a = a + bpîï = -2y = 2 (mod(l -?,,)2),

33 Finally by Theorem 5.1 we have BINOMIAL COEFFICIENTS AND JACOBI SUMS 463 4,(9,10) = - (y) (mod 77), 4,(8,8) = -(^) (mod 77), 4,(9,9) ee- J^j (mod 77), so (14.5) [^][\^/[\f^-y=\(a-bfñ) (mod 77). But from (14.3) and (14.4) we have (14.6) Ua + bfä\) =0 (mod 77). Hence from (14.5) and (14.6) we have 7)(S) r)/ I 2 J =a (mod 77), and so appealing to (2.2), we obtain Theorem Ifp = 11/+ 1 is prime and a is defined uniquely by 4 p = a >2, a EE 2 (mod 11), we then have m/(î/) =fl (mod^)' This is equivalent to Jacobi's result [19] 1 "-/!3/!4/!5/!9/T (m d/,)- Example. With/7 = 89, so that/ = 8, ü = -9, 7 = 5, we have /24U48\ //32\ _(64)(72) _ 79 _ I 8 JI24J/ I 16 j = 22 =TT = "9 (mod89)- 15. e = 12. There are eight representative binomial coefficients of order 12, namely, We show that all the binomial coefficients in (15.1) can be determined from the lower order binomial coefficients (\ff),(%) and (4^). We begin by determining (3^) in terms of (\ff) modulo p. Let 5 be a prime ideal divisor of p in Q(Çn) and define g and Xn as m 3- Then it is known (see, for example, Whiteman [44, p. 61]) that 42(3,3) = -a + 7/ where p = a2 + >2, a = 1 (mod 4).

34 464 R. H. HUDSON AND K. S. WILLIAMS Appealing to Whiteman's cyclotomic numbers of order 12 [44] we have (15.2) 42(l,2) = (-l)/cx,2(3)42(2,4), where c is given by (15.3) (-\)f (-1) /+! ifa = 1 (mod3), 7EE0(mod3), if a EE2(mod3), >EE0(mod3), (-!)'/ if 6= 1 (mod3),flee0(mod3), /; (-l)'i if 7 = 2 (mod 3), a = 0 (mod 3). Now 3 is clearly a quadratic residue of p so that x*q) = (-1)7 if > = 0 (mod 3) and X4(3) = (-l)f+i if a = 0 (mod 3) (see, for example, [22, p. 24]). Taking conjugates on both sides of (15.2) we have where Appealing to Theorem 5.1, we have 42(10,11) =ec42(10,8), + 1 if 7 = 0(mod3), -1 ifûee0(mod3). 10/ (-i)/+'i10/) = -r;;i (mod5). 4/ Finally, as 42(3,3) = -a + 7/ ee 0 (mod P) we have, using (2.2), (15.4) where (15.5) (-D1 (-D/+1 (-i)v«\/+1 3/Mv M if a ee 1 (mod 3), 7 EE0(mod3), if a ee 2 (mod 3), 7 EE0(mod3), if >EE 1 (mod 3), a =0(mod3), (-1); 7/a if > ee 2 (mod3), a = 0 (mod3). We now show that the 7 remaining binomial coefficients of order 12 may be determined in terms of lower order binomial coefficients. Corollary relates (5/) and (ff) modulo p. However, Corollary gives a simpler congruence, namely, (15.6) 5/ / 8/ 4/ (mod p ). Corollary gives the congruence 6/\.. / 6/ /)= U

35 BINOMIAL COEFFICIENTS AND JACOBI SUMS 465 Corollary gives the congruence (15.7) (y) =(2"-»/3)a(y) EEe(2^-')/3)2 6/j {modp) Appealing to (2.2) and Lemma 2.1 (g = 2, h = 1, k = 9) we have (y)/(7)-<-'>i7ra-m/m <-» Thus, using (4.3), we obtain (4/)H2/)(r/)/(3/)-"-'i2/)(26//)/(3/)<-^ Now using (15.4) and (15.7) we have (15.8) (TlHv) (m d/,)' Next using Lemma 2.1 (g = 7, = 3, k = 4) we have so, using (15.5) and (15.8), we obtain (15.9) (vh'^'lv) (m d/,)- Again appealing to Lemma 2.1 (g = 6, A = 3, k = 5), we have Using (15.9) we have ($/( H-0/(f/) *-') (15.10) (25/Hr'(2/) {m0dp)- Finally, appealing to Lemma 2.1 (g = 4, h = 2, & = 5) and using (2.2), we have ($/&)-<-<)/(!HS)/(S) *-'> Using (4.3), (15.9) and (15.10) we have so that after cancellation we obtain (15.11) (27/)^(-1)/2<'")/3e(3/) (m d/,)-

36 466 R H HUDSON AND K S. WILLIAMS Combining (15.4) and (15.11) and appealing to (4.5), Corollary 4.1.1, and Theorems 6.1, 7.1 and 9.2, we have Theorem Let p = 12/+ 1 = a2 + >2 = x2 + 3y2 be a prime with a = 1 (mod 4), x ee 1 (mod 3), and let 4p = A2 + 21B2 with A = 1 (mod 3). Then we have the following congruences modulo p: (y)--*- (3;h- (i)-**- (5;m(6/h- /5/\ 2x lip U/j " 0 ' \2f) where 0 is given by (15.5) and $ by 4> 2(-\)fe2<t,a. (^)=2(-l)'fc,, 1 ify = 0(mod3), (x + 3y)/(x-3y) ify = 1 (mod3), (x - 3y)/ (x + 3y) ify = 2 (mod3). Example. For p = 97, formulas (15.4) (15.11) and Theorem 14.1 can be easily checked from the following brief table of values. (See Table 2.) 16. e = 13. Since a,(43(l,3)) = 43(3,9) = 43(1,3), 43(1,3) is an integer of the field g(//26 + 6/ÎT ). Zee [15, p. 263] has shown that (16.1) 43(1,3) = ^lx + w{\3 +/íi/y/26 + 6/Í3" + t>^26-6/ï3 )), where (x, u, v, w) is a solution of the system (16.2) 16/7 = x2 + 26«2 + 26t)2 + 13w2, x = 9 (mod 13), ^xw = 3v 4uv 3w We prove Theorem If p = 13/+ 1 is prime then and 4/\ x 3(x2- \3w2)w f}=-^ + ~tt-, n \ (mod/7) / / 2 8(xw + 13«ti) lf\ _ x 3(x2- \3w2)w 2fj 2 8(xw + \3uv) (mod/7), where (x, u, v, w) is any solution of (16.2). Proof. The ring of integers of Q(Ç]3) is a unique factorization domain (see, for example, [27]). Let 77 be a prime dividing p in Q(${3). By Theorem 5.1 and (2.2), we have (16.3) (yu(1 /)=--UlO,12) (mod 77).