CHAPTER IV. RESIDUE OF (Up~i-l)/P MODULO P. N. H. Abel1 asked if there are primes p and integers a for which (1) a'-'sl (modp2), Ka<p. C. G. J. Jacobi2 noted that, for pg37, (1) holds only when p = ll, a=3 or 9; p=29, a = 14; p=37, a = 18. Cf. Thibault31 of Ch. VI. G. Eisenstein3 noted that, for p a prime, the function fl«=( has the properties (2) gu.sg«+g., (mod p), where s = (p+l)/2,. . ., p — 1. All solutions of (1) are included in as u+puqu, Q<u<p. E. Desmarest4 noted that (1) holds for p=487, a = 10, and stated that p=3 and p=487 are the only primes <1000 for which 10 is a solution. J. J. Sylvester5 stated that, if p, r are distinct primes, p>2, then qr is congruent modulo p to a sum of fractions with the successive denominators p— 1, . . . , 2, 1 and (as corrected) with numerators the repeated cycle of the positive integers ^r congruent modulo r to 1/p, 2/p, . . ., r/p. Thus, for r-5, According as p=4fc+l or 4fc—1, q2 is congruent to 2 222 2 p-3 p-4 p-7 p-8 p-11 '' ___2____2____2____2 2 p-2 p-3 p-6 p-7 p-10,'" [the signs were given + erroneously]. For any p, >Jour. ftir Math., 3, 1828, 212; Oeuvres, 1, 1881, 619. */Md., 301-2; Werke, 6, 238-9; Canon Arithmeticus, Berlin, 1839, Introd., xxxiv. •Berlin Berichte, 1850, 41. «Th^orie des nombres, 1852, 295. •Comptes Rendus Paris, 52, 1861, 161, 212, 307, 817; Phil. Mag., 21, 1861,136; Coll. Math. Papers, II, 229-235, 241, 262-3.