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HISTORY OF THE THBOBY OF NUMBERS. [CHAP, iv modulo p2 of o>* is a root, that (±z2) of a?!2 is a second root, that (±s3) of %z2 is a third root, etc., until the root ±x. is reached, where s = (p-l)/2 The remaining roots are p2-^^!,. . ., s). He proved that P+I (xi...z,)2=(-l) 2 (modp2). Hence xi. . .xt===*=l if p=4n+l. W. Meissner39 wrote hm for the residue <pw of V"4 modulo pm. When h varies from 1 to p~l, we get p-1 roots /im of x*~l=l (mod p"). The product of the roots given by fc = l,. . ., (p-l)/2, is ss(-i)" or (-l)V (mod pm), according as p=4n-l or 4n+l, where 2 is the number of pairs of integers <p/2 whose product is s= -1 (mod p), and a* is the smaller of the two roots of afe — l (mod p). No number <p which belongs to one of the exponents 2, 3, 4, 6, modulo p, can be a root of x*-^l (mod p2). A root of the latter is given for each prime p<300, and a root modulo p3 for each p<20G; also the exponent to which each root belongs. N. Nielsen40 noted that, if we select 2r distinct integers a,, 6, (s = 1, . . . ,r) from I, . . ., p — 1, such that a8+b8 = p, then Am Proof is given of various results by Lerch,21 also of simple relations between ga and Bernoullian numbers, and of the final formula by Plana,6 here attributed to Euler.41 H. S. Vandiver42 proved^ that there are not fewer than [Vp] and not more than p — (1 + V2p— 5)/2 incongruent least positive residues of 1, 2*-x,..., (p-l)^\ modulo p2. N. Nielsen43 noted that, if a is not divisible by the odd prime p, ft — 1 (P-3>/2 1 '-IB.(a>-2'-1-l) (modp), -(-l)n"1^n+~-l (modp2), W. Mdssner" pave various expressions for g2 A. Otfrardiri46 found all primes p<2000, including those of the form 2n~~ I, for which g2 is symmetrical when written to the base 2. H. S. Vandiver46 proved that g2=0 (mod p2) if and only if l+| + J+... + -^0 (modp2). 3 5 P~2 He gave various expressions for (nfc—l)/m. '•SitnmfltHbor. Berlin Math, Orsoll., 13, 1014, 96-107. *°Ann. BC. lY'Colc norm. Bup., (3), 31, 1914, 171-9. "Kulw, Institutioncs Calculi Diff., 1755, 406. Proof, Math. Quest. Educ. Times, 48, 1888, 48. "Hull. Amer. Math. Soc., 22, 1915, 61-7. **Ovcr«iKt Danske Vidensk. Selsk. Forhandlinger, 1915, 518-9, 177-180; cf. Lerch's81 N. "Mitt. Math. Gesell. Hamburg, 5, 1915, 172-6, 180. *&Nouv. Ann. Math., (4), 17, 1917, 102-8. '"Annals of Math., 18, 1917, 112.