CHAPTER III. 
FERMAT'S AND WILSON'S THEOREMS, GENERALIZATIONS AND 
CONVERSES; SYMMETRIC FUNCTIONS OF 
1,2 P-\ MODULO P. 
Fermat's and Wilson's Theorems; Immediate Generalizations. 
The Chinese^ seem to have known as early as 500 B. C. that 2^—2 is 
divisible by the prime p. This fact was rediscovered by P. de Fermat^ 
while investigating perfect numbers. Shortly afterwards, Fermat^ stated 
that he had a proof of the more general fact now known as Fermat's theorem: 
If p is any prime and x is any integer not divisible by p, then x^~^ — 1 is 
divisible by p. 
G. W. Leibniz^ (1646-1716) left a manuscript giving a proof of Fermat's 
theorem. Let p be a prime and set x = a+6+c+. . .. Then each multi- 
nominal coefficient appearing in the expansion of x^ — 2a^ is divisible by p. 
Take a = 6 = c=...=l. Thus a;^ — a: is divisible by p for every integer x. 
G. Vacca^ called attention to this proof by Leibniz. 
Vacca^ cited manuscripts of Leibniz in the Hannover Library showing 
that he proved Fermat's theorem before 1683 and that he knew the theorem 
now known as Wilson's^^ theorem: If p is a prime, l + (p — 1)! is divisible 
by p. But Vacca did not explain an apparent obscurity in Leibniz's state- 
ment [cf. Mahnke'^]. 
D. Mahnke'' gave an extensive account of those results in the manuscripts 
of Leibniz in the Hannover Library which relate to Fermat's and Wilson's 
theorems. As early as January 1676 (p. 41) Leibniz concluded, from the 
expressions for the ^th triangular and yth. pyramidal numbers, that 
(2/+l)2/=2/'-2/=0 (mod 2), {y+2){y+l)y=y'-y=0 (mod 3), 
and similarly for moduU 5 and 7, whereas the corresponding formula for 
modulus 9 fails for y = 2, — thus forestalling the general formula by Lagrange.^* 
On September 12, 1680 (p. 49), Leibniz gave the formula now known as 
Newton's formula for the sum of like powers and noted (by incomplete 
induction) that all the coefficients except the first are divisible by the 
exponent p, when p is a prime, so that 
a''+h''+c''-{- . . . = {a+h+c+ . . .Y (mod p). 
Taking a = b= . . . =1, we obtain Fermat's theorem as above.'* That the 
binomial coefficients in (1 + 1)^ — 1 — 1 are divisible by the prime p was 
^G. Peano, Formulaire math., 3, Turin, 1901, p. 96. Jeans.^^" 
''Oeuvres de Fermat, Paris, 2, 1894, p. 198, 2°, letter to Mersenne, June (?), 1640; also p. 203, 
2; p. 209. 
"Oeuvres, 2, 209, letter to Frenicle de Bessy, Oct. 18, 1640; Opera Math., Tolosae, 1679, 163. 
*Leibnizens Math. Schriften, herausgegeben von G. J. Gerhardt, VII, 1863, 180-1, "nova 
algebrae promotio." 
»Bibliotheca math., (2), 8, 1894, 46-8. 
•Bolletino di BibUografia Storia Sc. Mat., 2, 1899, 113-6. 
'Bibliotheca math., (3), 13, 1912-3, 29-61. 
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