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Chap. Ill] SYMMETRIC FUNCTIONS MODULO Jp. 103 
powers of a primitive root /i of p in a rectangular table of t rows and r col- 
umns, where ir = p — 1. For p = 13, /i = 2, i = 4, the table 
is shown here. Let R range over the numbers in any 
column. Then Si2 and Sl/^R are divisible by p. If Ms 
even, Sl/i? is divisible by p^ as 1/1 + 1/8+1/12+1/5 = 
13^/120. For t = p — \, the theorem becomes the first one 
due to Wolstenholme.^^^ Generalizations are given at the end of the 
paper. 
N. Nielsen^^^ proved his^^^*" theorem and the final results of Glaisher.^^^ 
Nielsen^^^ proceeded as had Aubry^°^ and then proved 
(p-l)/2 ^_3 
S2„+i^0 (mod f), S r^O (mod p), l^n^ ^ . 
Then by Newton's identities we get Wilson's theorem and Nielsen's^"^ last 
result. 
E. Cahen^^^ stated Nielsen's^^^'' theorem. 
F. Irwin stated and E. B. Escott^^* proved that if Sj is the sum of the 
products J at a time of 1, 1/2, 1/3,. ., \/t, where ^= (p — 1)/2, then 2S2—Si^, 
etc., are divisible by the odd prime p. 
"iQversigt Danske Vidensk. Selsk. ForhandUnger, 1915, 171-180, 521. 
3i276id., 1916, 194-5. 
'i^Comptes Rendus Stances Soc. Math. France, 1916, 29. 
»"Ainer. Math. Monthly, 24, 1917, 471-2. 
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