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356                    HISTORY OF THE THEORY OF NUMBERS.           [CHAP, xin
E. Lebon93 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 Bibliothque de PInstitut.
H. W. Stager94 gave theorems on numbers which contain no factors of the form p(kp+l), where &>0 and p is a prime, and listed all such numbers <12230.
Lehmer95 listed the primes to ten million.
A. Ge*rardin96 discussed the finding of all primes between assigned limits
by use of stencils for 3, 5, 7, 11,___ He97 described his manuscript of
an auxiliary table permitting the factoring of numbers to 200 million. He980 gave a five-page table serving to factor numbers of the second million. Corresponding to each prime M 14867 is an entry P such that N=1000 000+P is divisible by M. If a value of P is not in the table, N is prune (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. Boulogne98 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 62=*= 1, while such a product is 6^=*= 1, where N = 6xy*=x+y or Qxyxy. A table of values of these N's up to k serves to find the composite numbers up to 6&. To apply this method to factor $N*= 1, seek an expression for N in one of the above three forms.
N. Alliston100 described a sieve (a modification of that by Eratosthenes) to determine the primes 4n+l and the primes 4n1.
H. W. Stager101 expressed each number < 12000 as a product of powers of primes, and for each odd prime factor gave the values >0 of k 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 5J12000.
In Ch. XVI are cited the tables of factors of a2+l by Euler,4'7 Escott,58 Cunningham63 and Woodall64; those of a2+/b2 (k = 1,..., 9) of Gauss13; those of yn+l, 2/4=t2, /1, 3v=fcj/*, 2ag, etc., of Cunningham.68'84'9. Concerning the sieve of Eratosthenes, see Noviomagus29 of Ch. I, Poretzky66 of Ch. V, Merlin139 and de Polignac305"7 of Ch. XVIII. Saint-Loup13 of Ch. XI, Reymond151 and Kempner152 of Ch. XIV, represented graphically the divisors of numbers, while Kulik134 gave a graphical determination of primes.
93L'interm6diaire des math., 19, 1912, 237.
"University of California Public, in Math., 1, 1912, No. 1, 1-26.
95List 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. franc, avanc. sc., 42, 1913, 2-8; 43, 1914, 26-8.
"/Md., 43, 1914, 17-26.
98aSphinx-Oedipe, se*rie spe"ciale, No. 1, Dec., 1913.
"L'enseignement math., 17, 1915, 93-9.   Cf. pp. 244-5 for remarks by GeVardin. IOOMath. Quest. Educat. Times, 28, 1915, 53.
J01A Sylow factor table of the first twelve thousand numbers.    Carnegie Inst. Wash. Pub. No. 151, 1916.