lg HlSTOBY OF THE THEORY OF NUMBERS. [CHAP. I Johann Christoph Heilbronner86 stated that the perfect numbers up to .4-107 are 6,28,496,8128,130816,2096128. " The fathers of the early church and many writers always held this number 6 in high esteem. God completed the creation in 6 days and since all things created by Him came out perfect, he wished the work of creation completed according to the number 6 as being a perfect number." L. Euler87 deduced from Format's theorem, which he here proved by use of the binomial theorem, the result* that, if m is a prime, 2m — 1, when composite, has no prime factors other than those of the form mn-fl. J. Landen88 noted that 196 is the least number 4zn, where x is prime, the sum of whose aliquot parts exceeds the number by 7. L. Euler89 gave a table of the prime factors of 2n—1 for n^ 37. C. N. de Winsheim90 noted that 247-l has the factor 2351, and stated that 2*-l is a prime for n=2, 3, 5, 7, 13, 17, 19, 31, composite for the remaining w<48, but was doubtful as to n=41, thus reducing the list of perfect numbers given by Euler83 by one or perhaps two. He suspected that n=41 leads to an imperfect number since it was excluded by the acute Mersenne,60 who gave instead 2^(2?.-l}_as the ninth perfect number. He remarked that the basis of Mersenne's asseBiiofflFdoubtless to be found in the stupendous genius of Mersenne which perhaps recognized more truths than he could demonstrate. He discussed the error of Hansch84 that 2n—1 is a prime if n is a prime ^ 79. G. W. Kraft91 considered perfect numbers AP, where P is a prime [not dividing A]. Thus a(P+l)=2AP, where a is the sum of all the divisors of A. Hence a/(2A—a) equals the prune P, Let 2A— a = l, a property holding for A=2m. Then P=2m+1-l and the resulting numbers are of Euclid's type. L. Euler,92 in a letter to Goldbach, October 28, 1752, stated that he knew only seven perfect numbers, viz., 2P~1(2P—1) for p = 2, 3, 5, 7, 13, 17, 19, and was uncertain whether 231-1 is prime or not (a factor is necessarily of the form 64n-fl and none are <2000). MHistoria matheseos universae. Accedit recensio elementorum compendiorum et operum math, atque historia arithmetices ad nostra tempora, Lipsiae, 1742, 755-6. There is a 63-page list of arithmetics of the 16th century. 87Novi Comm. Ac. Petrop., 1, 1747-8, 20; Comm. Arith., I, 56, §39. *We may simplify the proof by using the fact that 2 belongs to an exponent e modulo p (p a prime) such that e divides p — 1. For, if p is a factor of 2OT-1, m is a multiple of e, whence e equals the prime m. Thus p-l =nm. If we take m>2, we see that n is even since p is odd and conclude with Fermat69 that, if m is an odd prime, 2m-l is divisible by no primes other than those of the form 2km+1. "Ladies' Diary, 1748, Question 305. The Diarian Repository, Collection of all the mathematical questions from the Ladies' Diary, 170^-1760, by a society of mathematicians, London, 1774, 509. Button's The Diarian Miscellany (from Ladies' Diary, 1704-1773), London, 1775, vol. 2, 271. Leybourn's Math. Quest, proposed in Ladies' D., 2, 1817, *»0puscula varii argumenti, Berlin, 2, 1750, 25; Comm. Arith 1 1849 104 »°Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, mem., 68-99. n/Wd., mem., 112-3. "Corresp. Math. Phys. (ed., Fuss), I, 1843, 590, 597-8.