THEOREMS ON QUADRA TIC PARTITIONS. 5. An2-polyhedra. Let 7rr(K) = 0 for r = 1,..., n - 1, where n > 2.

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1 60 MATHEMATICS: A. L. WHITEMAN PROC. N. A. S. 5. An2-polyhedra. Let 7rr(K) = 0 for r = 1,..., n - 1, where n > 2. In this case we may identify Fn+1 with'0 Hn(2) and bn+2 determines a homomorphism, (2) :Hn+2(2) -> Hn(2), which is the dual of" Sqn-2:H`(2) > HJ+2(2). The structure of In+2, as an extension of I,,+, by H,(2), is determined by 11(2). Thus Sn+2(K) is determined, up to a proper isomorphism, by the co-homology system H(K), or by the analogous system of homology groups,'2 in which 11(2) plays the part of Sqn-2* All our complexes will be simply connected CW-complexes, as defined in 5 of J. H. C. Whitehead, "Combinatorial Homotopy I," Bull. Am. Math. Soc., 55, (1949). This paper will be referred to as CH I. 2 In the light of this theorem a Jmcomplex, K, as defined in CH I, is seen to be one such that n:wr(k) H.(K), if t n < m, and 'm+j is onto, where K is the universal covering complex of K. This, and the other theorems stated here, will be proved in a paper which is to appear in the Annals of Mathematics. 3 An isomorphism will always mean an isomorphism onto. 4 Cf. Eilenberg, S., and MacLane, Saunders, "General Theory of Natural Equivalences," Trans. Am. Math. Soc., 58, (1945). 6 See Whitehead, J. H. C., "On Simply Connected, 4-Dimensional Polyhedra," Comm. Math. Helvetici, 22, (1949). This paper will be referred to as "SCP." 6 See equation (7.3) in Whitney, Hassler, "Relations Between the Second and Third Homotopy Groups of a Simply Connected Space," Ann. Math., 50, (1949). 7 Cf. Fox, R. H., "Homotopy Groups and Torus Homotopy Groups," -Ibid., 49, (1948). 8 Cf. Hirsch, G., "Sur le troisieme groupe d'homotopie des polyderes simplement connexe," C. R. Acad. Sci. Paris, 228, (1949), in case K is finite and without 2-dimensional torsion. Hirsch's representation of r3-13h4 can be obtained from Theorem 5 and Theorem 7 below. I In the forthcoming paper t1(m) is defined more generally and is shown to be natural. 10 See Whitehead, J. H. C., "The Homotopy Type of a Special Kind of Polyhedron," Ann. Soc. Polonaise Math., 21, (1949); also Whitehead, G. W., "On spaces with vanishing low-dimensional homotopy groups", Proc. Nat. Acad. Sci., 34, (1948). 11 Steenrod, N. E., "Products of Cocycles and Extensions of Mappings," Ann. Math., 48, (1947). 12 See a forthcoming paper by P. J. Hilton. THEOREMS ON QUADRA TIC PARTITIONS By ALBERT LEON WHITEMAN DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SOUTHERN CALIFORNIA Communicated by H. S. Vandiver, November 23, 1949 If p is a prime of the form 3f + 1, the diophantine equation 4p = a2 + 3b2 has a unique solution in a and b with a = 1(mod 3) and b 0(mod 3). About forty years ago von Schrutkal derived the formula a = (4), where Xt(n) is the Jacobstahl sum defined by

2 VOL. 36, 1950 MA THEMA TICS: A. L. WHITEMAN 61 4k(n) = h hkn (1) h=o )P P ( the symbol (h/p) denoting the quadratic character of h with respect to p. Recently, the following analogous result has been found by E. Lehmer.2 If p is a prime of the form 5f + 1, then the yalue of x in the pair of diophantine equations 16p = x2 + 50u2 + SOy w2 and xw - v 4uv - u2 is given by x = (4) with x 1 (mod 5). It is interesting to ask if a similar formula holds for a prime p of the form 7f + 1. The purpose of this note is to point out that this is not the case; we offer in its place the result stated in Theorem 2. Our method is,based on the theory of cyclotomy. Let g be a primitive root of a prime p of the form ef + 1. A number N, prime to p, is congruent to a power of g: N ges + h(mod p), O< s _ f-1, 0 < h.e-1. For fixed h and k, 0 < h, k < e - 1, the cyclotomic number (h, k) is defined as the number of sets of values of s and t, each chosen from 0, 1, f - 1 for which the congruence ges+hh 1 + get+k (mod p) holds. and The numbers (h, k) satisfy, among others, the following properties :' (h, k) = (k, h) = (e-k, h-k), e odd, (2) e-1 E (k, h) =f-nk, k Op l,... =, e -1, (3) h =O where nk = 1 if k = 0 and nk = 0 if k 4 0. Let,B denote a primitive eth root of unity. The numbers prime to p may be distributed into e classes according to the residue classes (mod e) of their indices to the base g. We accordingly define the following extension of the Legendre symbol: (= ind N (4) for an integer N prime to p. For an integer N divisible by p, we put (N/p)e = 0. If N is an eth residue of p, that is, if the congruence xe _ N(mod p), p / N, is solvable, then (N/p)e = 1, regardless of the value of g. Otherwise, the value of (N/p)e depends on the primitive root taken as base.

3 62 MATHEMATICS: A. L. WHITEMAN PROC. N. A. S. For any pair of integers m, n define the cyclotomic function R(m, n) by means of the equation p-2 R(m, n) = E m ind t-(m + n) ind (1 + t). (5) From the theory of cyclotomy4 we have the following fundamental property of R(m, n). If no one of the integers m, n and m + n is divisible by e, then R(m, n)r(-m, -n) = p. (6) Furthermore, the value of R(-m, -n) may be derived from the value of R(m, n) by replacing (3 by -'1. In this note5 we shall investigate the cases e = 3 and e = 7. Assume first that e = 3. In this case, the formulas in equation (2) may be expressed schematically by means of the matrix A B C B C D (7) C D B in which the element in the hth row and kth column, h, k = 0, 1, 2, represents the value of the cyclotomic number (h, k). In terms of this notation the equations in equation (3) become A +B+ C =f- 1,B+ C+D =f. (8) Let f(k) denote the number of distinct solutions of the congruence x2 + x k(mod p). (9) Then the sum,f(a), where a runs over the (p - 1)/3 cubic residues of a p, is equal to the number of values of x, 1 _ x < p - 1, for which x2 + x is a cubic residue of p, that is, for which (x/p)3((x + 1)/p)s = 1. In view of equation (7), this number is equal to (0, 0) + (1, 2) + (2, 1) = A + 2D. It follows that P-1 Ef(s3) = f(o) + 3 E f(a) = 2 + 3(A + 2D). (10) s=o a In equation (5) take m = n = 1, and = co, a primitive cube root of unity. Using equation (4), we get R(1,1) = E,ind -2 ind (1+) - E1 +t t=1t=p Putting R(1, 1) = r + sw + t.w2 we deduce from equations (6) and (7) that 4p = a2 + 3b2, where

4 VOL. 36, 1950 MA THEMA TICS: A. L. WHITEMAN 63 and a=2r-s-t, b = s-t, (11) r = A + 2D, s =3B, t =3C. (12) Combining equations (8), (10), (11) and (12), we obtain p-1 a = -P+2+3r = -p + f(s3). (13) s=o Let r,, be a number not divisible by p such that (rv/p)3 = c', v = O, 1, 2. Then we may show in a similar fashion that 1p- p- \ b E-i fr s) -)) -Effr (14) Note that equation (13) implies that the sign of a is such that the congruence a = 1(mod 3) is satisfied, whereas the sign of b in equation (14) depends on the primitive root g employed. We have thus proved the following theorem. THEOREM 1. The values of a and b in the equation 4p = a2 + 3b2, where p = 3f + 1 is a prime, are given by equations (13) and (14), where the sign of a is determined by the condition a = 1 (mod 3), and f(k) denotes the number of distinct solutions of the congruence (9). As an application we observe that the formulaf(k) = 1 + ((4k + i)/p) lead at once to the theorem of von Schrutka stated in the introduction. We turn now to the case e = 7. In this case the formulas in equation (2) yield the matrix A B C D E F G B G HI J K H C H F K L L I D I K E J L J (15) E J L J D I K F K L L I C H G HI J K H B in which the letter in the hth row and kth column, h, k = O, 1, 2..., 6, represents the value of (h, k). In terms of this notation the equations in equation (3) reduce to A +B+ C+D+E+ F+G = f- 1, B + G + 2H + I + J + K = f, (16) C+ F+H+I+K+2L =f, D + E + I + K + 2J + L =f.

5 .s4 MATHEMATICS: A. L. WHITEMAN PROC. N. A. S. Let g(k) denote the number of distinct solutions of the congruence X3 + x2 _ k(mod p). (17) The sum Eg(a), where a runs over the (p - 1)/7 incongruent seventha power residues with respect to the modulus p, is equal to the number of values of x, 1 < x p - 1, for which x3 + X2 is a seventh-power residue of p, that is, for which (X/p)7 ((x + 1)/p)7 = 1. In view of equation (15), this number is equal to (0, 0) + (1, 5) + (2, 3) + (3, 1) + (4, 6) + (5, 4) + (6, 2) = A + 3I + 3K. It follows that P-1 E g(s7) = g() + 7 g(a) 2 + 7(A K). (18) 's=o a In equation (5) take m = 1, n = 2, and,b = 0, a primitive seventh root of unity. Making use of equation (4), we obtain R(1, 2) = E oind t -3 ind (1 + t) 1 (1 + t (19) 1=1 1=1 7 p 7 It follows from equations (15) and (19) that R(1, 2) may be written in the form R(1, 2) = r + s( ) + t( ), where r = A + 3I + 3K, s = B + C+ E + H+ J + K+L, (20) t = D+F+G+H+I+ J+L. Applying equation (6), we get also 4p = a2 + 7b2, where a = 2r -s-t, b = s - t. (21) Combining equations (16), (18), (20) and (21) we obtain P-1 3a = -P r = -p + E g(s7). (22) s =o Let r,, be a number not divisible by p such that (rv/p)7 = 0, v = 0, 1 2,..., 6. Then we may show in a similar fashion that ( P-i P-1 b = - E g(r1s7) - E g(r6s7). (23) The formula in equation (22) implies that the sign of a is such that the congruence a = 5 (mod 7) is satisfied; the sign of b, however, depends on the choice of the primitive root g. We have therefore proved the following theorem. THEOREM 2. The values of a and b in the equation 4p = a2 + 7b2, where p = 7f + 1 is a prime, are given by equations (22) and (23),where the sign

6 VOL. 36, 1950 MA THEMA TICS: A. L. WHITEMAN 65 of a is determined by the condition a =-5(mod 7), and g(k) denotes the number of distinct solutions of the congruence (17). The following result due to Dickson6 may be used as a criterion for determining the value of g(k). Let p be a prime >3, and let R = -4a3-27b2 be the discriminant of the cubic congruence x3+ax + O(modp). This congruence has a single integral root if and only if R is a quadratic non-residue of p; it has three distinct integral roots if and only if R is the residue of a square 81C2 $ 0 and e = 1/2(- b + c\/-3) is the residue of the cube of a number u + vv/-3 in which u and v are integers; it has no integral root if and only if R is a quadratic residue and e is not the residue of such a cube. In order to apply this theorem it is only necessary to observe that for k # 0, g(k) is the number of solutions of the congruence x8- kx - = O(mod p), where kk = 1(mod p). Unfortunately, Dickson's criterion does not lead to a simple formula expressing the value of a in Theorem 2 in terms of the Jacobstahl sum defined in equation (1). 1 Von Schrutka L., J. Reine Angew. Math., 140, (1911). Von Schrutka's result has been rediscovered by S. Chowla, Proc. Lahore Philos. Soc., 7, 2 pp. (1945). 2 Lehmer, E., Bull. Am. Math. Soc., 55, (1949), Abstract No Bachmann, P., Die Lehre von der Kreisteilung, 2nd ed., B. G. Teubner, 1921, pp Ibid., p Bachmann's formula (3) is equivalent to our formula (6). 6 For a treatment of the case e = 5, see a forthcoming paper by the author in Duke Math. J. 6 Dickson, L. E., Bull. Am. Math. Soc., 13, 1-8 (1906).

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