Chap. XVIII] MISCELLANEOUS RESULTS ON PRIMES. 437 
E. Dormoy^^^ noted that, if 2, 3, . . ., r, s, t, u are the primes in natural 
order, all primes (and no others) <u^ are given by 
2-3. . .stm-\-Dtat-\-tCtD,a,+tsCtC,Drar-\- . . . 
-\-tsr. . .7-5C,C,Cr. . .C,D,as+ts. . .5'3C,Cs. . .C3, 
where Ct is found from the quotients obtained in finding the g. c. d. of t 
and 2-3 .. .rs by a rule which if applied to four quotients a, h, c, d consists in 
forming in turn 1, p = cZc+l, p&-|-^, {ph+d)a+p = Ct. Further, A = ^C^=i=l, 
the sign being + or — according as there is an odd or an even number of 
operations in the g. c. d. process. 
C. de Polignac^^^" wrote p„ for the nth prime and discussed the express- 
ibility of all numbers, under a specified limit and divisible by no one of 
pi, . . ., pn-i, in the form 
{P2,P3,--->Pn-hPn) + {p3,P4,- ■ ■,PnPl)+. . . + (Pl, . . . , Pn-l) , 
where (a, 6, ... ) denotes ± a^b^ .... For example, every number < 53 and 
divisible by neither 2 nor 3 is of the form ±3 "±2^. 
J. J. Sylvester^^^ proved that if m is prime to i and not less than n, the 
product (m+i)(m+2i) . . . (w+m) is divisible by some prime >n. 
A. A. Markow^^*^ found a fragment in a manuscript by Tchebychef 
aiding him to prove the latter's result that if // is the greatest prime divi- 
sor of (1+2^) (1+4^) . . .(1+4A^^), then fx/N increases without limit with 
N (cf. Hermite, Cours, ed. 4, 1891, 197). 
J. Iwanow-^^ generalized the preceding theorem as follows: If fx is the 
greatest prime divisor of (A + 1^). . .(A+L^), then fi/L increases without 
limit with L. 
C. Stormer^^^ concluded the existence of an infinitude of primes from 
Tchebychef's^^^resultandusedthelattertoprovethat2'(i — l)(t — 2) . . .(i—n) 
is neither real nor purely imaginary if n is any integer 5^ 3, and i = \/ — 1. 
E. Landau^' (pp. 559-564) discussed the topics in the last three papers. 
Braun^^" proved that the (n+l)th prime is the only root Xt^I oi 
where ai = 2, 02, . . , «„ are the first n primes. 
C. Isenkrahe^^^ expressed a prime in terms of the preceding primes. 
R. Le Vavasseur^^° noted that all primes between p„ and p\+i, where p„ 
is the nth prime, are given by Sjii qiWiPJpi (mod P,J, where Pn = PiP2 
• • • Pn and WiPJpi=l (mod p,). 
"^Comptes Rendus Paris, 63, 1866, 178-181. 
284<»Comptes Rendus Paris, 104, 1887, 1688-90. 
285Messenger Math., 21, 1891-2, 1-19, 192. Math. Quest. Educ. Times, 56, 1892, 25. 
286Bull. Acad. Sc. St. Petersbourg, 3, 1895, 55-8. 
^^Ubid., 361-6. 
"sArchiv Math, og Natur., Kristiania, 24, 1901-2, No. 5. 
289Math. Annalen, 53, 1900, 42. 
290M6m. Ac. Sc. Toulouse, (10), 3, 1903, 36-8. 
