Download History of the Theory of Numbers, Volume I: Divisibility and by Leonard Eugene Dickson PDF
By Leonard Eugene Dickson
This 1st quantity within the sequence historical past of the idea of Numbers provides the fabric on the topic of the themes of divisibility and primality. This sequence is the paintings of a unique mathematician who taught on the college of Chicago for four many years and is well known for his many contributions to quantity idea and crew conception. 1919 variation.
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257 for which the composition of 2" -1 is in doubt. According to a "'Since these four values of n are included in the list by Lucas'" of the 28 values of n;:;;257 for which the composition of 2"-1 is unknown. Cf. Lucas"', p. 236. , 1, 1882, 241; 2, 1883, 230. Later, Lucas'" credited LeLaB8eur with these four cases as well as n=73 [Euler"] and n=79, 113, 233 [cf. ReuschlelO']. The last four cases were given by Lucas"" while the last three do not occur in the table (Lucas"', pp. 788-9) by LeLasseur of the proper divisors of 2"-1 for each odd n, n<79, and for a few larger composite n's.
A. Saverien94 repeated the remarks by Ens51 without reference. L. Euler 95 stated in a letter to Bernoulli that he had verified that 231 _1 is a prime by examining the primes up to 4633'9 which are contained in the possible forms 248n+1 and 248n+63 of divisors. L. Euler 96 gave a· prime factor of 2":1::: 1 for various values of n, but no new cases 2" -1 with n a prime. L. Euler,97 in a posthumous paper, proved that every even perfect number is of Euclid's type. Let a=2nb be perfect, where b is odd.
The P 3 120 and 672 are mentioned in the 1770 edition of Ozanam's79 Recreations, I, p. 35, and in Hutton's translation of Montucla's99 edition, I, p. 39. A. M. {3'Y .. , where a, {3, 'Y, ... are distinct odd primes, for m=3, n~8; m=4, n=3, 5; m =5, n=7. No new Pn were found. "'Oeuvres, 2, 1894, p. 247 (261), letter to Carcavi; Varia opera, p. 178; Precis des oeuvres math. de Fermat, par E. Brassinne, Toulouse, 1853, p. 150. 2, 1894, 255, letter to Mersenne, April 7, 1643. The editors (p. 256, note) explained the method of factoring probably used by Fermat.