Chap. VIII] 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; ax''+hy''-}-cz''=0, under Fermat's last theorem. 
Fermat^^^ stated that not every prime p divides one of the numbers 
a+1, a^+l,a^+l,. . .. For, if /c is the least value for which a^' — 1 is divis- 
ible by p and if k is odd, no term a^-{-l is divisible by p. But if k is even, 
^fc/2_|_2 jg divisible by p. 
Fermat^^^ stated that no prime 12n±l divides S'^+l, every prime 
12n=t5 divides certain S'^+l, no prime 10n±l divides 5""+!, every prime 
lOn^S divides certain 5"^+!, and intimated that he possessed a rule relating 
to all primes. See Lipschitz.^^*^ 
A. M. Legendre^^° obtained from a given congruence x"=ax'*~^+- ■ • 
(mod p), p SiTi odd prime, one having the same roots, but with no double 
roots. Express x^^"^'^^ in terms of the powers of a; with exponents <n, and 
equate the result to +1 and to —1 in 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. 
Hoen4 de Wronski^^^ developed {ni-\- . . .-\-nJ"', replaced each multi- 
nomial coefficient by unity, and denoted the result by A[ni+ . . .+nj\"*. 
Thus A[ni+n2f = ni^+nin2+n2^. SetiV„ = ni+ . . . +n„. Then (pp.65-9), 
(1) A[N^-nX-A[N^-nX={n,-n,)A[NT~'=0 (mod n,-n,). 
Let (ni. . .nS)m be the sum of the products of ni,. . ., n„ taken m at a 
time. Then (p. 143), if A[Nj' = l, 
(2) A[NJ={n,.. .nJ,A[N^Y-'-in,.. .nJ^AWJ-' 
+ in,.. .n^)sA[NJ-'- . . .+{-iy+\n,. . .nJ),A[Nj. 
He discussed (pp. 146-151) in an obscure manner the solution of Xi=X2 
(mod X), where the X's are polynomials in ^ of degree v. Set N^ = ni-\- . . . 
H-n„_2+np+ng. Let the negatives of Ui,..., n„_2, Up be the roots of 
P = Po-\-PiX+ . . .+P^^2^"~^-\-x"~^ = 0; the negatives of ni,..., n^-2, "riq 
the roots of Q = Qo+ . . .+x"~^ = 0. We may add fiX and ^2^ to the 
members of our congruence. It is stated that the new first member may 
be taken to be A[A^„— nj"", whence by (2) 
X,-\-^,X = P^_2A[N^-nr-'-P.-3A[N^-nX-^+ . . ., 
and the A's may be expressed in terms of the P's by (2). Similarly, 
^2+^2^ niay be expressed in terms of the Q's. By (1), X = nq—np = Q^_2 
— P„_2. Since P = 0, Q = have co — 2 roots in common, we have further 
conditions on the coefficients Pi, Qi. It is argued that w — 3 of the latter 
"^Oeuvres, 2, 209, letter to Frenicle, Oct. 18, 1640, 
i^Oeuvres, 2, 220, letter to Mersenne, June 15, 1641. 
i"M6m._Ac. Sc. Paris, 1785, 483. 
"^Introduction a la Philosophic des Math^matiques et Technie de I'Argorithmie, Paris, 1811. 
He used the Hebrew aleph for the A of this report. Cf. Wronski^^' of Ch. VII. 
