200 History of the Theory of Numbers. [Chap, vii 
and only if a""^ — 1 is not divisible by p^. If t is even, the least x for which 
a"+l = (mod p") is l^p""'"". 
]\I. Cipolla^°^gave a historical report on congruences (especially binomial), 
primitive roots, exponents, indices (in Peano's symboUsm). 
K. P. Nordlund^°^ proved by use of Fermat's theorem that, if rij, . . ., n,. 
are distinct odd primes, no one dividing a, then A^" = ni"*' . . . n^*"' divides 
a'-l, where A;=0(iV)/2^-^ 
R. D. Carmichael^°^ proved that the maximum indicator of any odd 
number is even; that of a number, whose least prime factor is of the form 
4ZH-1, is a multiple of 4; that of p(2p — 1) is a multiple of 4 if p and 2p — 1 
are odd primes. 
A. Cunningham^^° gave a table of the values of v, where {p — l)/v is the 
exponent to which 2 belongs modulo p"< 10000, the omitted values of p 
being those for which i' = 1 or 2 and hence are immediately distinguished 
by the quadratic character of 2 (extension of his Binary Canon^^). A list 
is given of errata in the table by Reuschle.^^ An announcement is made of 
the manuscript of tables of the exponents to which 3, 5, 6, 7, 10, 11, 12 
belong modulo p"< 10000, and the least positive and negative primitive 
roots of each prime < 10000 [now in type and extended in manuscript to 
p"< 22000]. 
A. Cunningham^ ^^ defined the sub-Haupt-exponent ^i of a base q to 
modulus m = q°-°y]Q (where 770 is prime to q, and ao^O) to be the exponent to 
which q belongs modulo r^o- Similarly, let ^2 be the exponent to which q 
belongs modulo 771, where ^i = 5'''i?i; etc. Then the ^'s are the successive 
sub-Haupt-exponents, and the train ends with ^,.+1 = 1, corresponding to 
77;. = 1 . His table I gives these ^k for bases g = 2, 3, 5 and for various moduli 
including the primes < 100. 
Paul Epstein^ ^^ desired a function ^{m), called the Haupt-exponent for 
modulus m, such that a'''^'"^ = 1 (mod m) for every integer a prime to m and 
such that this will not hold for an exponent <\p{m). Thus \f/{m) is merely 
Cauchy's^^ maximum indicator. Although reference is made to Lucas, ^^ 
who gave the correct value of 4^(ni), Epstein's formula requires modification 
when m = 4 or 8 since it then gives \p = l, whereas \p = 2. The number 
x(w, m) of roots of x''= 1 (mod m) is 2dodi . . .d„ if m is divisible by 4 and if 
H is odd, but is di . . . ci„ in the remaining cases, where, for m = 2'*°pi*i . . .pn'"'*, 
di is the g. c. d. of jj, and 4>{pi°-^), and do the g. c. d. of fx and 2°""^, when 
ao>l. The number of integers belonging to the exponent /x = pV-- 
modulo m is 
\x{m, p°)-x(m, p°-^)[ \x{m, q^)-x{m, (f-^)\. . .. 
"^Revue de Math. (Peano), Turin, 8, 1905, 89-117. 
"8G6teborgs Kungl. Vetenskaps-Handlingar, (4), 7-8, 1905, 12-14. 
"»Amer. Math. Monthly, 13, 1906, 110. 
"OQuar. Jour. Math., 37, 1906, 122-145. Manuscript announced in Mess. Math., 33, 1903-4, 
145-155 (with list of errata in earUer tables); British Assoc. Report, 1904, 443; I'inter- 
m^diaire des math., 16, 1909, 240; 17, 1910, 71. CI. Cunningham."^ 
"iProc. London Math. Soc, 5, 1907, 237-274. 
"^Archiv Math. Phys., (3), 12, 1907, 134-150. 
