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Full text of "History Of The Theory Of Numbers - I"

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CHAP. xvi] FACTOBS OP a*±b*. 389 L. E. Dickson and E. B. Escott69 discussed the divisibility of pn/8-l by d(pn/d— 1), where d is & divisor of n, and d of d. R. D. Carmichael70 proved that if PSa—RSa is divisible by 5a and we set Q = (Pa -#")/ {a(P-K)}, then Q/8 is an integer if and only if a is divisible by the least integer e for which Pe—R* is divisible by each prune factor of a not dividing P—R, and 5 is a divisor of Q. Proof for the case R = 1 had been given by E. B. Escott71. A. Cunningham72 tabulated the factors of ym±l for y = 2, 3, 5, 7, 12. K. J. Sanjana73 considered the factors of Sanjana730 applied his method to prove the statement of M. Kannan that 2046-1=11-19-31-61'251421-3001-261451-64008001 -3994611390415801 •4199436993616201. L. E. Dickson74 factored nn— 1 for various values of n. R. D. Carmichael76 employed the methods of Dickson66 to obtain generalizations. Let Qn(a, /3) be the homogeneous form of Fn(a). Let n=IIpt0», where the p's are distinct primes, and let c be a divisor of n and a multiple of pi\ If a, ft are relatively prime, the g. c. d. of 5 = an/Pi— /3n/p> and Qc(a, 0) is 1 or PI and at most one Qc(a, /3) contains the factor pL when 6 contains p*] if Pi>2 divides 6, at most one Qc(a, jS) contains pi, and no one of them contains p*. If a, (3 are relatively prime and c=mp!% where m>l and m is prime to plt then Qc(a, /3) is divisible by pi if and only if ax=]Sz (mod p^ holds for z = w, but not for 0<z<?n; in all other cases Q== 1 (mod m). If a, j3 are relatively prime, Qc(a, j3), and hence also ac—/3c, has a prime factor not dividing a*— /3*(s<c), except in the cases (i) c = 2, ft = 1, a = 2* - 1 ; (ii) Qc(a, 0) = p = greatest prime factor 6f c, and an/p^ftn/v (modp); (iii)Qc(a,/3) = l. E. Miot76 noted that LeLasseur's23 formula is the case m=n=l of Welsch (p. 213) stated that the latter is no more general than the case fc = 0, which follows from the known formula for the product of two sums of two squares. A. Cunningham77 noted the decomposition into primes: 277+l = 343-617-683-78233-35532364099. «»L'interm6diaire des math., 1906, 87; 1908, 135; 18, 1911, 200. Cf. Dickson.87 70Amer. Math. Monthly, 14, 1907, 8-9. "Ibid., 13, 1906, 155-6. "Report British Assoc., 78, 1908, 615-6. 73Proc. Edinburgh Math. Soc., 26, 1908, 67-86; corrections, 28, 1909-10, viii. 78aJour. Indian Math. Club, 1, 1909, 212. "Messenger Math., 38, 1908, 14-32, and Dickaonu8~» of Ch. XIV. "Amer. Math. Monthly, 16, 1909, 153-9. 7flL'interme*diaire des math., 17, 1910, 102. "Report British Assoc. for 1910, 529; Proc. London Math. Soc., (2), 8, 1910, xiii.