426 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvm The common difference is divisible by each prime ^n, and by n itself if n is a prime not in the series. H. Brocard1890 gave several sets of five consecutive odd integers, four of which are primes. Lionnet169Z> had asked if the number of such sets is unlimited. G. Lemaire170 noted that 7+30n and 107-f 30n (n = 0, 1,..., 5) are all primes; also 7+150n and 47+210n (n = 0,..., 6). E. B. Escott171 found conditions that o-f210n (n = 0, 1,..., 9) be all primes and noted that the conditions are satisfied if a = 199. Devignot172 noted the prunes 47-1-21071, 71+2310n (n = 0, 1,..., 6). A. Martin173 gave numerous sets of primes in arithmetical progression. TESTS FOR PRIMALITY. The fact that n is a prime if and only if it divides l + (n—1)! was noted by Leibniz,7 Lagrange,18 Genty,24 Lebesgue,85 and Catalan,106 cited in Chapter III, where was discussed the converse of Fermat's theorem in furnishing a primality test. Tests by Lucas, etc., were noted in Ch. XVII. Further tests have been noted under Cipolla172 and176 Cole173 of Ch. I, Sardi273 of Ch. Ill, Lambert6 of Ch. VI, Zsigmondy79 of Ch. VII, Gegenbauer80-92 of Ch. X, Jolivald84 of Ch. XIII, Euler,26'43 Tchebychef,62 Schaffgotsch100 and Biddle141 of Ch. XIV, Hurwitz41 and Cipolla46 of Ch. XV. See also the papers by von Koch,238 Hayashi,239' 24° Andreoli,244 and Petrovitch246 of the next section. L. Euler177 gave a test for the primality of a number N=4m+l which ends with 3 or 7. Let R be the remainder on subtracting from 2N the next smaller square (5n)2 which ends with 5. To R add lOO(n-l), 100(n-3), 100(n — 5), .... If among R and these sums there occurs a single square, N is a prime or is divisible by this square. But if no square occurs or if two or more squares occur, N is composite. For example, if #=637, (5n)2 = 1225, # = 49; among 49, 649, 1049, 1249 occurs only the square 49; hence A1" is a prime or is divisible by 49 [# = 49-13]. W. L. Kraft178 noted that 6m+1 is a prime if m is of neither of the forms 6xy=±(x+y)'} 6m —1 is a prime if m^Qxy-}-x — y. A. S. de Montferrier179 noted that an odd number A is a prime if and only if A+k2 is not a square for k = 1, 2, ..., (A — 3)/2. M. A. Stern180 noted that n is a prime if and only if it occurs n — 1 times in the (n—l)th set, where the first set is 1, 2, 1; the second set, formed by inserting between any two terms of the first set their sum, is 1, 3, 2, 3,1; etc. "o«Nouv. Ann. Math., (3), 15, 1896, 389-90. *"*Nouv. Ann. Math., (3), 1, 1882, 33(6. l70L'interme'diaire des math., 16, 1909, 194-5. 171J&^.? 17, 1910, 285-6. ™Ibid., 45-6. 173School Science and Mathematics, 13, 1913, 793-7, 176Doubt as to the sufficiency of Cole's test has been expressed, Proc. London Math. Soc., (2). 16, 1917-8. I770pera postuma, I, 188-9 (about 1778). 178Nova Acta Acad. Petrop., 12, 1801, hist., p. 76, mem., p. 217. 179Corresp. Math. Phys. (ed., Quetelet), 5, 1829, 94-6. l80Jour. fur Math., 55, 1858, 202.