History Of The Theory Of Numbers - I

CHAP, xvi FERMAT NUMBERS Fn=22n+l. • 379
A. Cunningham470 noted that 3, 5, 6, 7, 10, 12 are primitive roots and 13, 15, 18, 21, 30 are quadratic residues of every prime Fn>5. He factored F4^+S+(FQF1F2F3)\
Thorold Gosset48 gave the two complex prime factors a=±= U of the known real factors of composite Fn, n = 5, 6, 9,11,12,18, 23, 36, 38.
J. C. Morehead49 verified by use of the criterion of Pepin19 with k = 3 that F7 is composite, a result stated by Klein.40
A. E. Western50 verified in the same way that F7 is composite. The work was done independently and found to agree with Morehead's.
J. C. Morehead51 found that F73 has the prime factor 275-5+l.
A. Cunningham52 considered hyper-even numbers
E0t n=2», Elt n=2*o,»,..., Er+1, n = 2*.«.
For ra odd, the residues modulo m of j^,0, J0r,i,... have a non-recurrent part and then a recurring cycle.
A. Cunningham53 gave tables of residues of E^ n, E2, „, Er, o, 33n and 55n 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 Fn, and of Fn modulo Fn+x.
Cunningham and H. J. Woodall54 gave material on possible factors of Fn.
A. Cunningham55 noted that, for every Fn>5, 2Fn~t2-(Fn-2)u2 algebraically, and expressed F5 and F6 in two ways in each of the forms a2+62, c2±2<22. He56 noted that Fn*+En3 is the algebraic product of n+2 factors, where En=22n, and that Mn=(Fn*+En*)/(Fn+En) is divisible by Mn.r. If n - m ^ 2, Fm4+Fn2 is composite.
A. Cunningham67 has considered the period of I/AT to base 2, where N is a product FmFm-i.. .Fm_r of Fermat numbers.
J. C. Morehead and A. E. Western58 verified by a very long computation that Fs is composite. Use was made of the test by Pepin19 with k = 3, which was proved to follow from the converse of Fermat's theorem.
P. Bachmann59 proved the tests by Pepin19 and Lucas.22
A. Cunningham60 noted that every Fn>5 can be represented by 4 quadratic forms of determinants =*=(?„, =t=2Gn, where Gn = F0Fi. . .Fn-i.
Bisman148 (of Ch. XIV) separated 16 cases in finding the factor 641 of F&.
470Math. Quest. Educ. Times, (2), 1, 1902, 108; 5, 1904, 71-2; 7, 1905, 72.
"Mess. Math., 34, 1905, 153-4. ^BuH. Amer. Math. Soc., 11, 1905, 543.
60Proc. 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. "Math. Quest. Educat. Times, (2), 12, 1907, 21-22, 28-31. »Ibid., (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.