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CHAP, ill]           GENERALIZATIONS OF FEBMAT'S THEOREM.                   89
Richard Saner190 proved that, if a, &, a— 6 are prime to k,
a*+a*>-l'b+a*-%*+ . . . +b*sl (mod k),           <P = v(k),
since a*"*"1— 6*+1sa— 6. Changing alternate signs to minus, we have a congruence valid if a, 6 are prime to k, and if a+b is not divisible by k. If p is an odd prime olividing a=p&,
is divisible by p, but not by p2.
A. Capelli191 showed that, if a, 6 are relatively prime,
where [re] is the greatest integer ^x.
M. Bauer192 proved that, if p is an odd prime and m = p° or 2p°, every integer £ relatively prime to m satisfies the congruence
(mod m),
where A?!,. . ., /bj denote the l=<t>(m) integers <m and prune to ra>2. If m is not 4, pa or 2pa, every integer re prime to m satisfies the congruence
(&'to<*-l)2=(x+kl)...(x+kd (mod m).
L. E. Dickson193 proved Moore's181 theorem by invariantive theory.
N. Nielsen194 proved that, if $(x) is a polynomial with integral coefficients 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
$(z) = <Kz) 6>p(s)+ S mp_, A. wf(x),    con(oO=z(o;-j-l) . .(s+n-1), «- 1
where <#>(o;) is a polynomial with integral coefficients, the ^4.8 are integers, p is the least positive integer for which p ! is divisible by m, and mp_3 is the least positive integer I for which s!Z is divisible by m. CL Borel and Drach.180
H. S. Vandiver195 proved that, if V ranges over a complete set of incon-gruent residues modulo m = pia . . .pjfc, while U ranges over those F's which are prime to m,
modulo m, where «. = (m/p*»)e, e =</>(p/0 .    For m = pa, the second congruence is due to Bauer.186- m
190Eine polynomische Verallgemeinerung dcs Fcrrnatachcn. Satzcs, Disa., Gieascn, 1905.
1MDritter Internal. Math. Kongress, Leipzig, 1905, 148-150.
1MArchiv Math. Phya., (3), 17, 1910, 252-3.    Cf. Bouniakowsky38 of Ch. XI.
"'Trans. Amer. Math. Soc., 12, 1911, 76; Madison Colloquium of the Amer. Math. Soc., 1914,
1MNieuw Archief voor Wiakunde, (2), 10, 1913, 100-6. i»Annalfl of Math., (2), 18, 1917, 119.