CHAP. VIII] SUM OF FOUR SQUARES. 297 



(Sn 3) 2 . The following theorems were proved by use of infinite series. 

 If N = 4n -f 1 and if H^ (or H 2 ) denotes the number of compositions of 

 4N as a sum of 4 odd squares of the same class (or not of same class), then 

 H l - |# 2 = X (N). As known, H l + H 2 = a(N). If N = 4n + 1 and if 

 of the partitions of 4N into 4 odd squares of which two are equal, P is the 

 number having the remaining two squares of the form (8n I) 2 and Q 

 the number for which they are of the form (8m 3) 2 , then P = Q if N is 

 not a square, while 



2 



Write S for (2p + I) 2 + (2q + I) 2 ; the number of representations of 

 Sn + 2 as S + (4r) 2 + (4s) 2 or S + (4r + 2) 2 + (4s + 2) 2 is respectively 



12{a(4n + 1) + x(4n + 1)}, 12{<r(4n + 1) - x (4n + 1)}; 



while there are \2a(4n + 3) representations 8n + 6 = S + (4r) 2 + (4s -f 2) 2 . 

 Let E(N) denote the excess of the number of divisors 4n + 1 of N 

 over the number of divisors 4n + 3; then E(N) is the number of primary 

 numbers of norm N. If n = 1 (mod 4), 



= E(l}E(2n - 1) - E(5)E(2n - 5) + #(9)#(2rc - 9) - 



<r(2m + 1) = ^(l)^(4w + 1) + E(5)E(4m - 3) + ^(9)J^(4w - 7) 



+ E(4m 



Call ^(n) the excess of the sum of the squares of the divisors 4m + 1 

 of n over the sum of the squares of the divisors 4m + 3; X(n) the sum of 

 the squares of the primary numbers of norm n. There are given many 

 formulas serving to evaluate x, <r, E, E 2 , X, whose values are tabulated 

 for arguments n ^ 100, with citation to longer tables. 



R. Lipschitz 77 found the number of sets of solutions of (} + & + & = 

 (mod p 7 ), where p is a prune, and applied the result to find all integral 

 quaternions with a given norm and hence the solutions of m = UK He 

 discussed the real and rational automorphs of x\ + x\ + x\. 



S. Re"alis 78 concluded from pq = a 2 + + 5 2 three sets of fractional 

 expressions for p and q in terms of a, , 6 and new parameters, but ad- 

 mitted that he was unable to utilize them to prove Bachet's theorem. 



A. Puchta 79 repeated Gauss' 17 derivation of Euler's formula (1). To 

 interpret (1), use the four-dimensional regular body bounded by 5 tetra- 

 hedra and having as vertices 5 equidistant points P,-. There exists a point 

 such that OP X , -, OP 4 are perpendicular lines, while the "planes" 

 through and any three of PI, , P 4 are perpendicular. We may take 

 to be the point with the coordinates Xi = (ai + a z + a 3 + a 4 ) /2, etc., 



77 Untersuchungen iiber die Summen von Quadraten, Bonn, 1886. French transl. by J. 



Molk, Jour, de Math., (4), 2, 1886, 393-439. 



78 Jour, de math. e"lem., (2), 10, 1886, 89-91. 



79 Sitzungsber. Akad. Wisa. Wien (Math.), 96, II, 1887, 110. 



