356 History of the Theory of Numbers. [Chap, xni 
E. Lebon'' stated that he constructed in 1911 a table of residues p, p' 
permitting the rapid factorization of numbers to 100 million, the manuscript 
being in the Bibliotheque de I'lnstitut. 
H. W. Stager^ gave theorems on numbers which contain no factors of 
the form p{kp-\-l), where k>0 and p is a prime, and listed all such numbers 
< 12230. 
Lehmer'^ listed the primes to ten million. 
A. G^rardin^^ discussed the finding of all primes between assigned limits 
by use of stencils for 3, 5, 7, 11,. . .. He^^ described his manuscript of 
an auxiliary table permitting the factoring of numbers to 200 million. He^^" 
gave a five-page table serving to factor numbers of the second million. Cor- 
responding to each prime M^ 14867 is an entr\' P such that A^ = 1 000 000+P 
is diWsible by M. If a value of P is not in the table, A^ is prime (the P's 
range up to 28719 and are not in their natural order). By a simple division 
one obtains the least odd number in any million which is divisible by the 
given prime M^ 14867. 
C. Boulogne^^ made use of lists of residues modulis 30 and 300. 
H. E. Hansen^^ gave an impracticable method of forming a table of 
primes based on the fact that all composite numbers prime to 6 are products 
of two numbers 6x± 1, while such a product is QN=^ 1, where N = 6xy='X+y 
or Qxy—x — y. A table of values of these A^'s up to k serves to find the com- 
posite numbers up to 6A-. To apply this method to factor 6N=^ 1, seek an 
expression for A^ in one of the above three forms. 
N. AUiston^°° described a sieve (a modification of that by Eratosthenes) 
to determine the primes 4n+l and the primes An — l. 
H. W. Stager^"^ expressed each number < 12000 as a product of powers 
of primes, and for each odd prime factor gave the values >0 of A: for all 
divisors of the form p{kp-{-l). The table thus gives a list of numbers which 
include the numbers of Sylow subgroups of a group of order ^ 12000. 
In Ch. XVI are cited the tables of factors of a^+1 by Euler,^' Escott,^^ 
Cunningham^^ and WoodalP; those of a--\-k- (^* = 1, . . . , 9) of Gauss"; those 
of ?/" + l, 2/^±2, y'=t. 1, x^zti/^, 2«=tg, etc., of Cunningham.^^- ^"^ Concern- 
ing the sieve of Eratosthenes, see No\'iomagus-^ of Ch..I, Poretzkj^^ of Ch. 
V, :MerUn"^ and de Polignac^*^"-^ of Ch. XVIII. Saint-Loup" of Ch. XI, 
Re>Tnond^^^ and Kempner^^^ of Ch. XIV, represented graphically the divi- 
sors of numbers, while Kulik^^ gave a graphical determination of primes. 
»»L'interin6diaire des math., 19, 1912, 237. 
•♦University of California Public, in Math., 1, 1912, No. 1, 1-26. 
•*LiHt of prime numbers from 1 to 10,006,721. Carnegie Inst. Wash. Pub. No. 165, 1914. The 
introduction gives data on the distribution of primes. 
"Math. Gazette, 7, 1913-4, 192-3. 
•'Assoc, frang. avanc. sc, 42, 1913, 2-8; 43, 1914, 26-8. 
»«/bw/., 43, 1914, 17-26. 
••"^Sphinx-Oedipe, s^rie sp4ciale. No. 1, Dec, 1913. 
••L'enseignement math., 17, 1915, 93-9. Cf. pp. 244-5 for remarks by G^rardin. 
"""Math. Quest. Educat. Times, 28, 1915, 53. 
'"A Sylow factor table of the first twelve thousand numbers. Carnegie Inst. Wash. Pub. No. 
151, 1916. 
