Chap. XV] FeRMAT NUMBERS F„ = 2^" + l. 379 
A. Cunningham^^" noted that 3, 5, 6, 7, 10, 12 are primitive roots and 
13, 15, 18, 21, 30 are quadratic residues of every prime F„>5. He fac- 
tored F4' + S+iFoFiF2Fz)\ 
Thorold Gosset^^ gave the two complex prime factors a± 6i of the known 
real factors of composite F„, n = 5, 6, 9, 11, 12, 18, 23, 36, 38. 
J. C. Morehead'*^ verified by use of the criterion of Pepin^^ with k = S 
that 7^7 is composite, a result stated by Klein.^° 
A. E. Western^" verified in the same way that Fy is composite. The 
work was done independently and found to agree with Morehead's. 
J. C. Morehead^^ found that Fy^ has the prime factor 2^^-5+1. 
A. Cunningham^^ considered hyper-even numbers 
Eo, „ = 2", E,,n = 2^0. n, . . . , E^+,, „ = 2^r, n. 
For m odd, the residues modulo m of Er,o, -E'r, i,- • • have a non-recurrent 
part and then a recurring cycle. 
A. Cunningham^^ gave tables of residues of Ei^ „, E2, n, Er, 0, 3^" and 5^" 
for the n's forming the first cycle for each prime modulus < 100 and for 
certain larger primes. A hyper-exponential number is like a hyper-even 
number, but with base q in place of 2. He discussed the quadratic, quartic 
and octic residue character of a prime modulo F^, and of F„ modulo Fn+x- 
Cunningham and H. J. WoodalP^ gave material on possible factors of F^. 
A. Cunningham^^ noted that, for every F„>5, 2Fn = t^ — (Fn — 2)u^ 
algebraically, and expressed F5 and Fq in two ways in each of the forms 
a^+6^ c^±2d2. He^*^ noted that Fn^+E^^ is the algebraic product of 
n-\-2 factors, where ^n = 2^", and that M„ = (F„H^„^)/(F„-fE„) is divisible 
by Mn-r- If n — m^2, F„i'^-\-FJ is composite. 
A. Cunningham^^ has considered the period of l/N to base 2, where N 
is a product FmFm-i . ■ -Fm-r of Fermat numbers. 
J. C. Morehead and A. E. Western^^ verified by a very long computation 
that Fg is composite. Use was made of the test by Pepin^^ with A; = 3, which 
was proved to follow from the converse of Fermat's theorem. 
P. Bachmann^^ proved the tests by Pepin^^ and Lucas. ^^ 
A. Cunningham*^^ noted that every Fn>5 can be represented by 4 
quadratic forms of determinants =^Gn, =*=2(r„, where Gn = FoFi. . .Fn-i- 
Bisman^'^^ (of Ch. XIV) separated 16 cases in finding the factor 641 of F^. 
^^'^Math. Quest. Educ. Times, (2), 1, 1902, 108; 5, 1904, 71-2; 7, 1905, 72. 
"Mess. Math., 34, 1905, 153-4. "BuU. Amer. Math. Soc, 11, 1905, 543. 
"Proc. Lond. Math. Soc, (2), 3, 1905, xxi. 
"Bull. Amer. Math. Soc, 12, 1906, 449; Annals of Math., (2), 10, 1908-9, 99. French transl. in 
Sphinx-Oedipe, Nancy, 1911, 49. ^^Report British Assoc. Adv. Sc, 1906, 485-6. 
"Proc London Math. Soc, (2), 5, 1907, 237-274. 
"Messenger of Math., 37, 1907-8, 65-83. 
55Math. Quest. Educat. Times, (2), 12, 1907, 21-22, 28-31. 
^HhU., (2), 14, 1908, 28; (2), 8, 1905, 35-6. 
"Math. Gazette, 4, 1908, 263. 
"Bull. Amer. Math. Soc, 16, 1909, 1-6. French transl., Sphinx-Oedipe, 1911, 50-55. 
"Niedere Zahlentheorie, II, 1910, 93-95. 
'"Math. Quest. Educat. Times, (2), 20, 1911, 75, 97-98. 
