Chap. XVIII] GoLDBACH's THEOREM. 421 
E. Miot^i^ stated that x^- 2999a; +2248541 is a prime for 1460^0;^ 1539. 
G. Frobenius"^ proved that the value of x^+xy+py^ is a prime if <p^, 
that of 2x^+py^ (y odd) if <p(2p+l), that of x'^+2py^ (x odd) if <p(p-\-2), 
and noted cases in which an indefinite form x^-\-xy—qy^ is a prime. 
Levy^^ examined x^—x — 1. He"^ considered f{x) = ax^-{-ahx-{-c, where 
a, h, c are integers, 0^a<4. Giving to x the values 0, 1, 2, . . ., we get a 
set of integers such that, for every n exceeding a certain value, f{n) is 
either prime or admits a prime factor which divides a number f(p), where 
p<n. For example, if for f{x) =a;^ — a:+41 we grant that /(O), /(I), /(2), 
/(3) and /(4) are primes, we can conclude that f{x) is prime for x^^O. 
Likewise when 41 is replaced by 11 or 17. Again, 2a;^ — 2a:+19 and 3x^ — 3a; 
+23 give successions of 18 and 22 primes respectively. Bouniakowsky^^ 
of Ch. XI considered polynomials which represent an infinitude of primes. 
Braun^^" proved that there exists no quotient of two polynomials such 
that the greatest integer contained in its numerical value is a prime for all 
integral values > A; of the variable. 
Goldbach's Empirical Theorem: Every Even Integer is a Sum of 
Two Primes. 
Chr. Goldbach^^° conjectured that every number N which is a sum of two 
primes is a sum of as many primes including unity as one wishes (up to N), 
and that every number >2 is a sum of three primes. 
L. Euler^^^ remarked that the first conjecture can be confirmed from an 
observation previously communicated to him by Goldbach that every even 
number is a sum of two primes. Euler expressed his belief in the last state- 
ment, though he could not prove it. From it would follow that, if n is 
even, n, n — 2, n — 4:,. . . are the sums of two primes and hence n a sum of 
3, 4, 5, . . . primes. 
R. Descartes^^^ stated that every even number is a sum of 1, 2 or 3 
primes. 
E. Waring^^^ stated Goldbach's theorem and added that every odd 
number is either a prime or is a suiri of three primes. 
L. Euler^^^ stated without profbf that every number of the form 4n+2 
is a sum of two primes each of the form 4A;+1, and verified this for 4n+2 
^110. 
"OL'intermediaire des math., 19, 1912, 36. [From X2+X+41 by setting X=x-1500.] 
"iSitz. Ak. Wiss. Berlin, 1912, 966-980. 
"^Bull. Soc. Math. France, 1911, Comptes Rendus des Seances. Extract in Sphinx-Oedipe, 
9, 1914, 6-7. 
i^oCorresp. Math. Phys. (ed., P. H. Fuss), 1, 1843, p. 127 and footnote; letter to Euler, June 7, 
1742. 
i"/&id., p. 135; letter to Goldbach, June 30, 1742. Cited by G. Enestrom, Bull. Bibl. Storia Sc. 
Mat. e Fis., 18, 1885, 468. 
^^'Posthumous manuscript, Oeuvres, 10, 298. 
i23Meditationes Algebraicae, 1770, 217; ed. 3, 1782, 379. The theorem was ascribed to Waring 
by O. Terquem, Nouv. Ann. Math., 18, 1859, Bull. Bibl. Hist., p. 2; by E. Catalan, Bull. 
Bibl. Storia Sc. Mat. e Fis., 18, 1885, 467; and by Lucas, Th^orie des Nombres, 1891, 353. 
i2^Acta Acad. Petrop., 4, II, 1780 (1775), 38; Comm. Arith. Coll., 2, 1849, 135. 
