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14 HlSTOBY OF THE THEORY OF NUMBERS. [CHAP. I He stated that while perfect numbers end with 6 or 28, the proof by Bungus42 does not show that they end alternately with 6 and 28, since Bungus included imperfect as well as perfect numbers. The numbers 130816 and 2096128, cited as perfect by Puteanus,53 are abundant. After giving a table of the expanded form of 2n for n=0, 1,..., 100, Broscius (p. 130, seq.) gave a table of the prime divisors of 2n— 1 (n = 1,. .., 100), but showing no prune factor when n is any one of the primes, other than 11 and 23, less than 100. For r&=ll, the factors are 23, 89; for n = 23, the factor 47 is given. Thus omitting unity, there remain only 23 numbers out of the first hundred which can possibly generate perfect numbers. Contrary to Cardan,27 but in accord with Bungus,42 there is (p. 135) no perfect number between 104 and 105. Of Bungus' 24 numbers, only 10 are perfect (pp. 135-140): those with 1, 2, 3, 4, 8, 10, 12, 18, 19, 22 digits, and given by 2»-1(2n-l) for n-2, 3, 5, 7, 13, 17, 19, 29, 31, 37, respectively. The pri-mality of the last three was taken on the authority of unnamed predecessors. There are only 21 abundant numbers between 10 and 100, and all of them are even; the only odd abundant number <1000 is 945, the sum of whose aliquot divisors is 975 (p. 146). The statement by Lucas, The*orie des nombres, 1, Paris, 1891, p. 380, Ex. 5, that 33-5-79 [deficient] is the smallest abundant number is probably a misprint for 945 = 33-5-7. This error is repeated in Encyclopedic Sc. Math., I, 3, Fas. 1, p. 56. Johann Jacob Heinlin62 (1588-1660) stated that the only perfect numbers <4-107 are 6, 28, 496, 8128, 130816, 2096128, 33550336, and that all perfect numbers end alternately in 6 and 8. Andrea Tacquet63 (Antwerp, 1612-1660) stated (p. 86) that Euclid's rule gives all perfect numbers. Referring to the 11 numbers given as perfect by Mersenne,60 Tacquet said that the reason why not more have been found so far is the greatness of the numbers 2n — 1 and the vast labor of testing their primality. Frenicle64 stated in 1657 that Euclid's formula gives all the even perfect numbers, and that the odd perfect numbers, if such exist, are of the form pk2, where p is a prime of the form 4n+l [cf. Euler98]. Frans van Schooten65 (the younger, 1615-1660) proposed to Fermat that he prove or disprove the existence of perfect numbers not of Euclid's type. Joh. A. Leuneschlos66 remarked that the infinite multitude of numbers contains only ten perfect numbers; he who will find ten others will know "Jolt. Jacob! Heinlim, Synopsis Math, praecipuas totius math.... Tubingae, 1653. Synopsis Math. Universalis, ed. Ill, Tubingae, 1679, p. 6. English translation of last by Venterus Mandey, London, 1709, p. 5, "Arithmeticae Theoria et Praxis, Lovanii, 1656 and 1682 (same paging), [1664, 1704], His opera math., Antwerpiae, 1669, does not contain the Arithmetic. "Correspondence of Chr. Huygens, No. 389; Oeuvres de Fermat, 3, Paris, 1896, p. 567. 8BOeuvres de Huygens, II, Correspondence, No. 378, letter from Schooten to J. Wallis, Mar. 18, 1658. Oeuvres de Fermat, 3, Paris, 1896, p. 558. "Mille de Quantitate Paradoxa Sive Admiranda, Heildelbergae, 1658, p. 11, XLVI, XLVII.