SQUARES THEOREM. Submitted by K. K. Azad. Absf ract. 0. Notation. As (n) = ~(k) ~ (- n sk), kot/s

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1 Far East J. Math. Sci. (FJMS) 3(6) (2001), AN ARITHMETIC PROOF OF JACOBI'S EIGHT SQUARES THEOREM KENNETH S. WILLIAMS ( Received July 21, 2001 ) Submitted by K. K. Azad Absf ract An elementary proof of Jacobi's eight squares theorem is g.lven. 0. Notation Let n and s denote positive integers. We let rs(n) denote the number of representations of n as the sum of s squares. We also let where d runs through the positive integers dividing n. If x is not a positive integer, we set os(x) = 0. We also define As (n) = ~(k) ~ (- n sk), kot/s where the summation is over all integers k satisfying 1.S k < n/s. Finally, we set 2000 Mathematics Subject Classification: 11E25. Key words and phrases: sums of eight squares, Jacobi's formula, identlty of Huard, Ou, Spearman and Wdliams. Research supported by Natural Sciences and Engineering Research Councd of Canada grant A O 2001 Pushpa Publishmg House

2 KENNETH S. WILLIAMS 1, ifsin, Fs(n) = 0, ifsrn. 1. Introduction The formula a (n) = 16(- 1)" (- 1ld d3 first appeared implicitly in the work of Jacobi [5], [6, Sections and explicitly in the work of Eisenstein [2], [3, p The standard arithmetic proof of (1) uses an elementary identity due to Liouville [8], see [lo, p. 4021, to show that the function on the right hand side of (1) satisfies the same recurrence relation as rs(n) with the same initial conditions so that the two functions are the same, see, for example, [lo, pp It is the purpose of this note to give a different arithmetic proof of (1). Our starting point is the following elementary identity due to Huard, Ou, Spearman and Williams [4], which is an extension of an identity of Liouville [7, p that Huard-OuSpearman-Williams Identity. Let f : z4 + C be such f(a,b,x,y)-f(x, y, a,b)=f(-a,-b,x, y)-f(x, y, -a, -b) for all integers a, b, x and y. Then where the sum on the left hand side of (2) is over all positive integers a, b, x, y satisfying ax + by = n, the inner sum on the right hand side is

3 JACOBI'S EIGHT SQUARES THEOREM over all positive integers x satisfying x < d, and the outer sum on the right hand side is over all positive integers d &viding n. The proof in [4, Section 21 of this identity is completely elementary as it only involves the rearrangement of terms in finite sums. The choice f(a, b, x, y) = xy in (2) yields the identity [4, eqn. (16)] (n) = 1 (503 (n) + (1-6n) o(n)), which originally appeared in a letter from Besge to Liouville [I]. The choice f(a, b, x, y) = (2a2 - b2) F4(x) yields the identity [4, Theorem 41 which is an extension of a result of Melfi [9, eqn. (ll)]. The choice four squares formula [4, Section 7] (Legendre-Jacobi-Kronecker symbol) gives Jacobi's r4 (n) = 8o(n) - 32o(n/4). Another arithmetic proof of (5) has been given by Spearman and Williams [ll]. Thus formulae (3), (4), (5) can all be proved by entirely elementary means. We now use these three results to give an arithmetic proof of (1). 2. Arithmetic Proof of Jacobi's Eight Squares Theorem We have.- - = Crq(k)q(n - k) = 2r4(n) + xr4(k)r4(n - k), as r4 (0) = 1. Appealing to (5), we obtain

4 1004 KENNETH S. WILLIAMS where Sl = o(k) o(n - k), S2 = o(k/4) o(n - k), k=l Clearly Sl = Al(n) and changing the summation variable in (10) from k to n - k shows that S3 = S2. Since the only terms in S2 and S4 which do not vanish are those for which 4 1 k, replacing k by 4k. in (9) and (1 l), we find that S2 = &(n) and S4 = Al(n/4). Appealing to (3) and (4) for the values of Al(n) and A4(n), and to (5) for the value of r4(n), we obtain from (6)-(11) Examining the three possibilities 2 1 n, 2 11 n and 4 1 n individually, we find that the right hand side of (12) is the same as the right hand side of (1). References [l] M. Besge, Extrait d'une lettre de M. Besge B M. Liouvllle. J. Math. Pures Appl. 7 (1862) [2] G. Eisensteln, Neue Theoreme der hoheren Arlthmetik, J. Relne Angew. Math. 35 (1847) G. Eisenstein, Mathematische Werke, Band I, Chelsea Publishing Company, New York (1989), [4] J. G. Huard, Z. M. Ou, B. K. Spearman and K. S. Williams, Elementary evaluation of certain convolution sums involwng divlsor functions, Proceedings of the Millennia1 Conference on Number Theory. University of Illinois (2000), to appear.

5 JACOBI'S EIGHT SQUARES THEOREM [5] C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Sumtibus Fratrum Borntraeger, Regiomonti, [6] C. G. J. Jacobi, Gesammelte Werke, Band I, Chelsea Pubkshing Company, New York (1969), [ J. Liouvllle, Sur quelques formules gknkrales qui peuvent Btre utdes dans la theorie des nombres (fifth article), J. Math. Pures Appl. 3 (1858) [8] J. Liouville, Sur quelques formules gbnkrales qui peuvent Btre utdes dans la theorie des nombres (twelfth article), J. Math. Pures Appl. 5 (1860), 1-8. [9] G. MeE, On some modular identities. Number Theory (K. Gyory, A. Pethij, and V. S~S, eds.), de Gruyter, Berlin (1998) (101 M. B. Nathanson, Elementary Methods in Number Theory, Spnnger, New York, [ll] B. K. Spearman and K. S. W~lliams, The simplest arithmetic proof of Jacobi's four squares theorem. Far East J. Math. Sci. (FJMS) 2 (2000), Centre for Research in Algebra and Number Theory School of Mathematics and Statistics Carleton University Ottawa, Ontario K1S 5B6 Canada william~math.carleton.ca