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 Biblioth£que 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 Qxy—x—y. 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 4n—1. 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/*, 2a±g, 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.