Chap. Ill] GENERALIZATIONS OF FeRMAT's ThEOREM. 89 
Richard Sauer^^'^ proved that, \i a,h, a — h are prime to k, 
a^+a^-^6+a^-V+ • • • +?)*'=1 (mod k), <P = (p{k), 
since a*""^^— 6*"^^=a— 6. Changing alternate signs to minus, we have a 
congruence valid if a, 6 are prime to k, and if 0+6 is not divisible by k. 
If p is an odd prime dividing a=F6, 
is divisible by p, but not by p^. 
A. Capelli^^^ showed that, if a, 6 are relatively prime. 
ah 
=[V]+[V]+i' 
where [x] is the greatest integer ^ x. 
M. Bauer^^^ proved that, if p is an odd prime and m = p" or 2p^, every 
integer x relatively prime to m satisfies the congruence 
(a;p-i _ l)p"- = (x+A^i) . . . {x-\-ki) (mod m), 
where /bi, . . ., ki denote the l=<l){m) integers <m and prime to m>2. If 
m is not 4, p" or 2p", every integer a: prime to m satisfies the congruence 
(x^^-^y^-iy^(x+ki) . . . (x+ki) (mod m). 
L. E. Dickson^^^ proved Moore's^^^ theorem by invariantive theory. 
N. Nielsen^^^ proved that, if ^{x) is a polynomial with integral coeffi- 
cients not having a common factor > 1, and if for every integral value of x 
the value of ^{x) is divisible by the positive integer m, then 
p-i 
^{x) = (f>{x) o)p{x)+ S rrip-s A, cos(x), o)n{x)=x{x-\-l) . .{x+n-1), 
8= 1 
where 4>(x) is a polynomial with integral coefficients, the Ag are integers, 
p is the least positive integer for which p ! is divisible by m, and mp_s is 
the least positive integer I for which s\l is divisible by m. Cf. Borel and 
Drach.i8° 
H. S. Vandiver^^^ proved that, if V ranges over a complete set of incon- 
gruent residues modulo m = pi . . .pl^, while U ranges over those F's 
which are prime to m, 
A; 
ll{x-V)^^tXx''^-xT"'% n(a;-C7)=Si,(a;P«-'-l)*'('")/(p»-^>, 
modulo w, where t^ = {mlpg^^y, e = </)(p/») . For w = p", the second congruence 
is due to Bauer.^^^' ^^^ 
""Eine polynomische Verallgemeinerung des Fermatschen Satzes, Diss., Giessen, 1905. 
»"Dritter Internat. Math. Kongress, Leipzig, 1905, 148-150. 
»»Archiv Math. Phys., (3), 17, 1910, 252-3. Cf. Bouniakowsky^s of Ch. XI. 
"'Trans. Amer. Math. Soc, 12, 1911, 76; Madison Colloquium of the Amer. Math. Soc, 1914, 
39-40. 
"<Nieuw Archief voor Wiskunde, (2), 10, 1913, 100-6. 
i»Annals of Math., (2), 18, 1917, 119. 
