426 History of the Theory of Numbers. [Chap, xviii 
The common difference is divisible by each prime ^n, and by n itself if n is 
a prime not in the series. 
H. Brocard^^^" gave several sets of five consecutive odd integers, four of 
which are primes. Lionnet^^^** had asked if the number of such sets is un- 
limited. 
G. Lemaire^^° noted that 7+30n and 107+30/1 (n = 0, 1,. . ., 5) are all 
primes; also 7 + 150/1 and 47+210m (n = 0, . . ., 6). 
E. B. Escott^^^ found conditions that a+210M (n = 0, 1,. . ., 9) be all 
primes and noted that the conditions are satisfied if a = 199. 
De\'ignot^" noted the primes 47+210«, 71+2310?i (n = 0, 1,. . ., 6). 
A. Martin^^^ gave numerous sets of primes in arithmetical progression. 
Tests for Primality. 
The fact that n is a prime if and only if it divides 1 + (n — 1) ! was noted 
by Leibniz/ Lagrange, ^^ Genty,-^ Lebesgue,^^ and Catalan/^^ cited in Chap- 
ter 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 Cipolla^" and^'^ Cole^" of Ch. I, Sardi"^ of 
Ch. Ill, Lambert^ of Ch. VI, Zsigmondv"^ of Ch. VII, Gegenbauer^o- »2 ^f 
Ch. X, Jolivald^ of Ch. XIII, Euler,^^^ Tchebychef,"^ Schaffgotsch^""^ and 
Biddle^^^ of Ch. XIV, Hurwitz" and CipoUa^^ of Ch. XV. See also the 
papers by von Koch,^^^ Hayashi,^^' ^° Andreoli,^"^ and Petrovitch^ of the 
next section. 
L. Euler^^^ gave a test for the primality of a number N = 4:m-\-l which 
ends with 3 or 7. Let R be the remainder on subtracting from 2N the next 
smaller square (5n)- which ends with 5. To R add 100(n — 1), 100(n — 3), 
100(n — 5), .... If among R and these sums there occurs a single square, 
iV is a prime or is divisible by this square. But if no square occurs or if 
two or more squares occur, A^ is composite. For example, if A = 637, 
(5n)2 = 1225, R = 49; among 49, 649, 1049, 1249 occurs only the square 49; 
hence A^ is a prime or is di\asible by 49 [A = 49- 13]. 
W. L. Kraft^"^ noted that Qin + 1 is a prime if m is of neither of the forms 
Qxy=^{x+y); 6?m — 1 is a prime if m9^6xy+x — y. 
A. S. de Montferrier^'^ noted that an odd number A is a prime if and 
only if A-\-k' is not a square for A: = 1, 2, . . ., (A — 3)/2. 
AI. A. Stern^^° 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. 
"'"Xouv. Ann. Math., (3), 15, 1896, 389-90. i696Nouv. Ann. Math., (3), 1, 1882, 336. 
""L'intcrraddiaire des math., 16, 1909, 194-5. 
"'Ibid., 17, 1910, 285-6. 
"*Jbid., 4.5-6. ^"School Science and Mathematics, 13, 1913, 793-7, 
'"*Doubt as to the suflBciency of Cole's test has been expressed, Proc. London Math. Soc, (2). 
16, 1917- 8. i"Opera postuma, I, 188-9 (about 1778). 
"»\ova Acta Acad. Petrop., 12, 1801, hist., p. 76, mem., p. 217. 
"•Corresp. Math. Phys. (ed., Quetelet), 5, 1829, 94-6. 
""Jour, fiir Math., 55, 1858, 202. 
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