200 HISTORY OF THE THEORY or NUMBERS. [CHAP. VII
and only if ap~l — 1 is not divisible by p2. If t is even, the least x for which a*+l=0 (mod pn) fe^"""1""-
M. Cipolla107gave a historical report on congruences (especially binomial), primitive roots, exponents, indices (in Peano's symbolism),
K. P. Nordlund108 proved by use of Fermat's theorem that, if n1? . . . , nr are distinct odd primes, no one dividing a, then N~n™1. . .nrmr divides a*-l, where k = <f>(N)/2r~l.
R. D. Carmichael109 proved that the maximum indicator of any odd number is even; that of a number, whose least prime factor is of the form 41+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. Cunningham110 gave a table of the values of i>, where (p — T)/v is the exponent to which 2 belongs modulo pn< 10000, the omitted values of p being those for which v~\ or 2 and hence are immediately distinguished by the quadratic character of 2 (extension of his Binary Canon95). A list is given of errata in the table by Reuschle.45 An announcement is made of the manuscript of tables of the exponents to which 3, 5, 6, 7, 10, 11, 12 belong modulo pn< 10000, and the least positive and negative primitive roots of each prime < 10000 [now in type and extended in manuscript to pn< 22000].
A. Cunningham111 defined the sub-Haupt-exponent & of a base q to modulus m=ga°7?0 (where I)Q is prime to q, and a0^0) to be the exponent to which q belongs modulo IJQ. Similarly, let £2 be the exponent to which q belongs modulo 77^ where ^i — q^i] etc. Then the £'s are the successive sub-Haupt-exponents, and the train ends with £r+1 = l, corresponding to r)r= 1. His table I gives these %k for bases # = 2, 3, 5 and for various moduli including the primes < 100.
Paul Epstein112 desired a function ^(w), called the Haupt-exponent for modulus m, such that a*w = 1 (mod m) for every integer a prime to ra and such that this will not hold for an exponent <\f/(m). Thus \l/(wi) is merely Cauchy's26 maximum indicator. Although reference is made to Lucas,73 who gave the correct value of \l/(m), Epstein's formula requires modification when m = 4 or 8 since it then gives ^ = 1, whereas \f/ = 2. The number x(wi, ju) of roots of #Ms= 1 (mod m) is 2dQ^ . . ,dn if m is divisible by 4 and if fj, is odd, but is di . . .dn in the remaining cases, where, for m = 2a°pla-1. . .pna», di is the g. c. d. of /-t and </>(pia«)> and <20 the g. c. d. of p and 2ao~2, when a0>l. The number of integers belonging to the exponent M = ?V---modulo m is
107Revue de Math. (Peano), Turin, 8, 1905, 89-117.
108G6teborgs Kungl. Vetenskaps-Handlingar, (4), 7-8, 1905, 12-14.
109Amer. Math. Monthly, 13, 1906, 110.
110Quar. Jour. Math., 3Z, 1906, 122-145. Manuscript announced in Mess. Math., 33, 1903-4, 145-155 (with list of errata in earlier tables); British Assoc. Report, 1904, 443; Tinter-m<§diaire des math., 16, 1909, 240; 17, 1910, 71. Cf. Cunningham.133
lllProc. London Math. Soc., 5, 1907, 237-274.
"'Archiv Math. Phys., (3), 12, 1907, 134-150.