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CHAPTER XV. FERMAT NUMBERS F«=22n+l. Fermat1 expressed his belief that every Fn is a prime, but admitted that he had no proof. Elsewhere2 he said that he regarded the theorem as certain. Later3 he implied that it may be proved by " descent." It appears that Frenicle de Bessy confirmed this conjectured theorem of Fer-mat's. On several occasions Fermat4 requested Frenicle to divulge his proof, promising important applications. In the last letter cited, Fermat raised the question if (2k)2m+l is always a prime except when divisible by an Fn. C. F. Gauss5 stated that Fermat affirmed (incorrectly) that the theorem is true. The opposite view was expressed by P. Mansion6 and R. Baltzer.7 F. M. Mersenne8 stated that every Fn is a prime. Chr. Goldbach9 called Euler's attention to Fermat's conjecture that Fn is always prime, and remarked that no Fn has a factor < 100; no two Fn have a common factor. L. Euler10 found that F5 = 232+1=641-6700417. Euler11 proved that if a and 6 are relatively prime, every factor of a2n+62>> is 2 or of the form 2n+1fc+land noted that consequently any factor of F5 has the form 64^+4^ k = 10 giving the factor 641. Eulerlla and N. Beguelin12<used the binary scale to find the factor 641 = l+27+29of F5. C. F. Gauss13 proved that a regular polygon of m sides can be constructed by ruler and compasses if m is a product of a power of 2 and distinct odd primes each of the form Fn, and stated that the construction is impossible if m is not such a product. This subject will be treated under Roots of Unity. Sebastiano Canterzani14 treated twenty cases, each with subdivisions depending on the final digits of possible factors, to find the factor 641 of F5, 'Oeuvres, 2,1894, p. 206, letter to Frenicle, Aug. (?) 1640; 2, 1894, p. 309, letter to Pascal, Aug. 29, 1654 (Fermat asked Pascal to undertake a proof of the proposition, Pascal, III, 232; IV, 1819, 384); proposed to Brouncker and Wallis, June 1658, Oeuvres, 2, p. 404 (French transl., 3, p. 316). Cf. C. Henry, Bull. Bibl. Storia So. Mat. e Fis., 12,1879, 500-1,716-7; on p. 717, 42... 1 should end with 7, ibid., 13,1880, 470; A. Genocchi, Atti Ac. Sc. Torino, 15,1879-80, 803. 'Oeuvres, 1, 1891, p. 131 (French transl., 3, 1896, p. 120). 'Oeuvres, 2, 433-4, letter to Carcavi, Aug., 1659. 40euvres, 2, 208, 212, letters from Fermat to Frenicle and Mersenne, Oct. 18 and Dec. 25,1640. BDisq. Arith., Art. 365. Cf. Werke, 2, 151, 159. Same view by Kliigel, Math. Worterbuch, 2, 1805, 211; 3, 1808,896. "Nouv. Corresp. Math., 5, 1879, 88, 122. 7Jour. far Math., 87, 1879, 172. 8Novarum Phyeico-Mathematicarum, Paris, 1647, 181. "Corresp. Math. Phys. (ed., Fuss), I, 1843, p. 10, letter of Dec. 1729; p. 20, May 22,1730; p. 32, July 1730. 10Comm. Ac. Petrop., 6, ad annos 1732-3 (1739), 103-7; Comm. Arith. Coll., 1, p. 2. "Novi Comm. Petrop., 1, 1747-8, p. 20 [9, 1762, p. 99]; Comm. Arith. Coll., 1, p. 55 [p. 357]. UflOpera postuma, I, 1862, 169-171 (about 1770). 1JNouv. Me"m. Ac. Berlin, anne~e 1777, 1779, 239. "Disq. Arith., 1801, Arts. 335-366; German transl. by Maser, 1889, pp. 397-448, 630-652. "Mem. 1st. Naz. Italiano, Bologna, Mat., 2, II, 1810, 459-469. 375