GOLDBACH'S THEOKEM. 421 E. Miot110 stated that a?-299ftc+2248541 is a prime for 1460ga^ 1539 G. Frobenius111 proved that the value of tf+xy+py* is a prime if <?>2" that of 2x*+py2 (y odd) if <p(2p+l), that of a?+2pjp (x odd) if <p(p+2)', and noted cases in which an indefinite form x^+xy-r-q^ is a prime L6vy15 examined z2-x -1. He112 considered /(x)= oz2+akc+c, where a, 6, c are integers, 0^ o< 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(ri) is either prime or admits a prime factor which divides a number /(p), where p< n. For example, if for f(x) =s2-z+41 we grant that /(O), /(I), /(2), /(3) and /(4) are primes, we can conclude that f(x) is prime for ±^40. Likewise when 41 is replaced by 11 or 17. Again, 2z2-2z+19 and 3z2-3x +23 give successions of 18 and 22 primes respectively. Bouniakowsky36 of Ch. XI considered polynomials which represent an infinitude of primes. Braun130 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 > k of the variable. GOLDS ACH'S EMPIRICAL THEOREM: EVERT EVEN INTEGER is A SUM OF Two PRIMES. Chr. Goldbach120 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. Euler121 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 statement, though he could not prove it. From it would follow that, if nis even, n, n — 2, n—4,.. . are the sums of two primes and hence n a sum of 3, 4, 5,... primes. R. Descartes122 stated that every even number is a sum of 1, 2 or 3 primes. E. Waring123 stated Goldbach's theorem and added that every odd number is either a prime or is a sum of three primes. L. Euler124 stated without proof that every number of the form 4n+2 is a sum of two primes each of the form 4&+1, and verified this for 4n+2 110L'interm6diaire des math., 19,1912, 36. [From X2+X+41 by setting X=z-1500.] mSitz. Ak. Wise. Berlin, 1912, 966-980. 112Bull. Soc. Math. France, 1911, Comptes Rendus des Stances. Extract in Sphinx-Oedipe, 9, 1914, 6-7. 120Corresp. Math. Phys. (ed., P. H. Fuss), 1, 1843, p. 127 and footnote; letter to Euler, June 7, 1742. ™Ibid., p. 135; letter to Goldbach, June 30,1742. Cited by G. Enestrom, Bull. Bibl. Storia Sc. Mat. e Fis., 18, 1885, 468. 1MPosthumous manuscript, Oeuvres, 10, 298. 123Meditationes 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, Th6orie des Nombres, 1891,353. 124Acta Acad. Petrop., 4, II, 1780 (1775), 38; Comm. Arith. Coll., 2, 1849, 135.