66 History of the Theory of Numbers. [Chap, hi 
J. A. Gninert^® considered the series 
K n] = n-- (^) (71-1)-+ Q (n-2r- . . ., 
to which Euler's (3) reduces for a = n, x = m, and proved that 
[m, n]=n\[m — l, n — l] + [w — 1, n]\ . 
This recursion formula gives 
[m,n] = (m = 0, l,...,n-l); [72, n]=n\ [cf. (2)], 
Any [m, n] is di\isible by n\. As by the proof of Lagrange,^^ [m, n] + ( — 1)" 
is di\isible by w + 1 if the latter is a prime >n. Again, 
which for x = 0, h=l, gives [m, m]=ml. 
W. G. Horner^" proved Euler's theorem by generaUzing Ivory's^^ method. 
If ri, . . . , r^ are the integers <m and prime to m, then riN, . . . , r^N have the 
r's as their residues modulo m. 
P. F. Verhulst^^ gave Euler's proof^^ in a sUghtly different form. 
F. T. Poselger^^ gave essentially Euler's^° first proof. 
G. L. Dirichlet^° derived Fermat's and Wilson's theorems from a com- 
mon source. Call m and n corresponding numbers if each is less than the 
prime p and if mn=a (mod p), where a is a fixed integer not di\dsible by p 
(thus generahzing Euler's-^ associated numbers). Each number 1, 2, . . ., 
p — 1 has (5ne and but one corresponding number. If a:"=a (mod p) has no 
integral solution, corresponding numbers are distinct and 
(p-l)!=a^-^)''2 (modp). 
But if A; is a positive integer <p such that ^'^=a (mod p), the second root 
is p — A', and the product of the numbers 1, . . ., p — 1, other than k and p—k, 
has the same residue as a^^~^^^^, whence 
(p-l)!=-a^-i^/2(j^Q^p) 
The case a = 1 leads to Wilson's theorem. By the latter, we have 
a(p-i)/2=±i (modp), 
"Math. Abhandlungen, Erste Sammlung, Altona, 1822, 67-93. Some of the results were 
quoted by Gnmert, Archiv Math. Phys., 32, 1859, 115-8. For an interpretation in 
factoring of [m, n], see Minetola'" of Ch. X. 
"Annals of Phil. (Mag. Chem. . . .), new series, 11, 1826, 81. 
>8Corresp. Math. Phys. (ed. Qu^telet), 3, 1827, 71. 
"Abhand. Ak. Wiss. Berhn (Math.), 1827, 21. 
"Jour, fiir Math., 3, 1828, 390; Werke, 1, 1889, 105. Dirichlet," §34. 
