CHAPTER XV. 
FERMAT NUMBERS F„ = 2'"+l. 
Fermat^ expressed his belief that every F^ is a prime, but admitted 
that he had no proof. Elsewhere^ he said that he regarded the theorem 
as certain. Later^ he impHed that it may be proved by "descent." It 
appears that Frenicle de Bessy confirmed this conjectured theorem of Fer- 
mat's. On several occasions Fermat^ requested Frenicle to divulge his proof, 
promising important applications. In the last letter cited, Fermat raised 
the question if (2/0)^""+ 1 is always a prime except when divisible by an F„. 
C. F. Gauss^ stated that Fermat affirmed (incorrectly) that the theorem 
is true. The opposite view was expressed by P. Mansion^ and R. Baltzer.'^ 
F. M. Mersenne^ stated that every F^ is a prime. Chr. Goldbach^ 
called Euler's attention to Fermat's conjecture that F„ is always prime, and 
remarked that no F^ has a factor < 100; no two F^ have a common factor. 
L. Euler^o found that 
^5 = 2^2+1 = 641-6700417. 
Euler^^ proved that if a and h are relatively prime, every factor of 
a^ +6^ is 2 or of the form 2"+^ A: + land noted that consequently any 
factor of F5 has the form 64fc + l, k = 10 giving the factor 641. 
Euler^^" and N. Beguelin^^ used the binary scale to find the factor 
641 = 1+2^+2^ of F5. 
C. F. Gauss^^ 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 F„, and stated that the construction is impossible 
if m is not such a product. This subject will be treated under Roots of Unity. 
Sebastiano Canterzani^^ treated twenty cases, each with subdivisions 
depending on the final digits of possible factors, to find the factor 641 of F^, 
iQeuvres, 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, BuU. Bibl. Storia So. Mat. e Fis., 12, 1879, 
500-1, 716-7; on p. 717, 42 ... 1 should end with 7, ihid., 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. 
^Oeuvres, 2, 208, 212, letters from Fermat to Frenicle and Mersenne, Oct. 18 and Dec. 25, 1640. 
^Disq. Arith., Art. 365. Cf. Werke, 2, 151, 159. Same view by Klugel, Math. Worterbuch, 
2, 1805, 211; 3, 1808, 896. 
«Nouv. Corresp. Math., 5, 1879, 88, 122. 
'Jour, fur Math., 87, 1879, 172. 
^Novarum Physico-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. 
"Comm. 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. CoU., 1, p. 55 [p. 357]. 
""Opera postuma, I, 1862, 169-171 (about 1770). 
"Nouv. Mem. Ac. Berlin, ann^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. 
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