CHAP, vni] MISCELLANEOUS RESULTS ON CONGRUENCES. 257 MISCELLANEOUS RESULTS ON CONGRUENCES. Linear congruences will be treated in Vol. 2 under linear diophantine equations, quadratic congruences in two or more variables, under sums of four squares; axn-i~byn+cznz=Q, under Fermat's last theorem. Fermat148 stated that not every prime p divides one of the numbers a+1, V2+l, a3+l, ---- For, if fc is the least value for which ah— I is divisible by p and if k is odd, no term ah+l is divisible by p. But i£'k is even, ak/2+l is divisible by p. Fermat149 stated that no prune 12n=*=l divides 3X+1, every prime 12n=»=5 divides certain 3X+1, no prime lOi=*=l divides 5*4-1, every prune 10n=*=3 divides certain 5X+1, and intimated that he possessed a rule relating to all primes. See Lipschitz.166 A. M. Legendre150 obtained from a given congruence xn=axn~~l+ . . -(mod p), p an odd prime, one having the same roots, but with no double roots. Express £(p~1)/2 in terms of the powers of x with exponents <n, and equate the result to + 1 and to — 1 hi turn. The g. c. d. of each and the given congruence is the required congruence. An exception arises if the proposed congruence is satisfied by 0, 1, . . ., p — 1. Hoe'ne* de Wronski151 developed (%+ . . . +nw)m, replaced each multinomial coefficient by unity, and denoted the result by A[UI-\- . . . -fnjm. ni2+n1n2+ri22. SetArw = n1+ . . . +nu. Then (pp.65-9), (1) AW.-njr-AWv-njr** (n^np)A[Nu]m'l^O (mod nfl-np). tim (2) Let (HI. . .nu)m be the sum of the products of nly . . ., nu taken m at a time. Then (p. 143), if A[NU]Q = 1, He discussed (pp. 146-151) in an obscure manner the solution of X^X2 (mod X), where the X's are polynomials in £ of degree v. Set Nu — n^ . . . *V Let the negatives of nif ..., nu-2, np be the roots of 4-...+ptt-aa-»-2+aj«-15asO; the negatives of ni,..., nw_2, nfl the roots of Q = Q0+ - - .H-oT^O. We may add ftX and f2Z to the members of our congruence. It is stated that the new first member may be taken to be A[NU— njm, whence by (2) and the A's may be expressed in terms of the P's by (2). Similarly, X2+tzX may be expressed in terms of the Q's. B.y (1), X = nq— np = Qu^2 — Pw_2. Since P = 0, Q = 0 have co — 2 roots in common, we have further conditions on the coefficients P;, Qt. It is argued that co— 3 of the latter M80euvres, 2, 209, letter to Frenicle, Oct. 18, 1640. ""Oeuvres, 2, 220, letter to Mersenne, June 15, 1641. w°Me*m. Ac. Sc. Paris, 1785, 483. "Introduction & la Philosophic des Math&natiques et Technie de 1'Argorithmie, Paris, 1811. TT* nflAfl t.^ft Hfthr^w alftnh for t.hft A nf this rfinart. C,i. Wrnnski169 of Oh. VTT.