HISTORY op THE THEORY OP NUMBERS. R. W. D. Christie63 stated that, for the recurring series defined by an+1=3an— an_!, 2m — I is a prime if and only if am—l is divisible by 2m — 1. The error of this test was pointed out by E. B. Escott.64 S. Re'alis64' noted that two of N consecutive terms of 7, 13, 25,..., 3(n2+n)+7,. . . are divisible by N if N is a prime 6ra-fl. C. E. Bickmore645 discussed factors of un in the final series of Catalan61. HeMc and others gave known formulas and properties of Pisano's series. R. Perrin65 employed ^ = ^-2+^-3, vQ = 3, ^ = 0, v2=2. Then vn is divisible by n if n is a prime. This was verified to be not true when n is composite for a wide range of values of n. The same subject was considered by E. Malo66 and E. B. Escott67 who noted that Perrin's test is incomplete. Several67" discussed the computation of Pisano's un for large n's. E. B. Escott676 computed 2l/un. E. Landau67' had evaluated 21/u2k in terms of the sum of Lambert's7 series of Ch. X, and Sl/^+i in relation to theta series. A. Tagiuri68 employed the series % = !, u2 = l, %=2,. . . of Leonardo and the generalization Uit U2,..., where I7n= Un_i+Un-.2, with Ui=a, U2=b both arbitrary. Writing e for (f-^-ab— b2, it is proved that {Un+&+(— iyUn^i}/Un is an integer independent of a, b, n; it equals u6+i+us~i. It is shown that ur is a multiple of ua if and only if r is a multiple of s. Tagiuri69 obtained analogous results for the series defined by Vn=hVn_i +lVn^2, and the particular series vn obtained by taking Vi~l, v2 = h. If h and I are relatively prime, vr is a multiple of va if and only if r is a multiple of 5. Let $(vt) be the number of terms of the series of t>'s which are ^ vt and prime to it; if h>l, *(^) is Euler's 0(i); but, if h = l, *fe)=0W+0(V2), the last term being zero if i is odd. If i and j are relatively prime, <£(%) Tagiuri70 proved that, for his series of t/s, the terms between vkp and ) are incongruent modulo vk if h> 1, and for h == 1 except for %p+i= vkjt+2-If IJL is not divisible by k and € is the least solution of I2k€z=l (mod fA), then tksst;M (mod t;*) if o;=ju (mod 4/ce). If /x is not divisible by /c, and k is odd, and ex is the least positive solution of P'sl (mod wjb), then vx^v^ (mod ^) if a?«M (m°d 2^€i). A. Emmerich71 proved that, in the series of Pisano, "Nature, 56, 1897, 10. "Math. Quest. Educat. Times, 3, 1903, 46; 4, 1903, 52 «<«Math. Quest. Educat. Times, 66, 1897, 82-3; cf. 72, 1900, 40, 71. Mft/6id., 71, 1899, 49-50. MCIUd., Ill; 4, 1903, 107-8; 9, 1906, 55-7. wL'intcrm6diaire des math., 6, 1899, 76-7. "Ibid., 7, 1900, 281, 312. 87L'interme"diaire des math., 8, 1901, 63-64. ""Ibid., 7, 1900, 172-7. 676/feid., 9, 1902, 43-4. 67CBull. Soc. Math. France, 27, 1899, 198-300. «8Periodico di Mat., 16, 1901, 1-12.