420 HlSTOBY OF THE THEORY OF NlJMBEES. [CHAP. XVIII R. D. Carmichael93 for pkn—1 (p an odd prime) and 2**3n—1. M. Bauer's94 paper was not available for report. POLYNOMIALS REPEESENTING NUMEROUS PRIMES. Chr. Goldbach100 noted that a polynomial /(re) cannot represent primes exclusively, since the constant term would be unity, whereas it is f(p) in f(x+p). L. Euler101 proved this by noting that, if /(a) = A, f(nA+a) is divisible by A. Euler102 noted that x2—re+41 is a prime for x = 1,..., 40. Euler103 noted that rc2+z+17 is a prime for rc = 0, 1,..., 15 and [error] 16; rc2+rc+41 is a prime for rc=0, 1, ..., 15. A. M. Legendre104 noted that rc2+rc+41 is a prime for rc = 0, 1,..., 39, that 2rc2+29 is a prime for re = 0, 1,..., 28, and gave a method of finding such functions. [Replacing x by x +1 in Euler's102 function, we get 32+z+41.] If /32+2(a+/3)rc-13rc2 is a square only when x= 0, and a and /3 are relatively prime, then a2+2a/3+14/32 is a prime or double a prime. He gave many such results. Chabert104a stated that 3n2+3n+l represents many primes for n small. G. Oltramare105 noted that x2+ax-\~b has no prime divisor g/x and hence is a prime when </i2, if a2—4b is a quadratic non-residue of each of the primes 2, 3,..., /*. The function rc2+aa;+(a2+163)/4 is suitable to represent a series of primes. Taking x—0, a=u/v, he stated that u2+16302 or its quotient by 4 gives more than 100 primes between 40 and 1763. H. LeLasseur106 verified that, for a prime A between 41 and 54000, rc2+z+A does not represent primes exclusively for rc = 0, 1,. .., A—2. E. B. Escott107 noted that rc2+rc+41 gives primes not only for re = 0,1, ..., 39, but also108 for rc= — 1, —2,..., —40. Hence, replacing x by re—40, we get re2—793+1601,. a prune for rc = 0, 1,..., 79. Several such functions are given. Escott109 examined values of A much exceeding 54000 in x2+x+A without finding a suitable -A>41. Legendre's104 first seven formulas for primes give composite numbers for a = 2, the eighth for a = 3, etc. Escott found that re3+a;2+17 is a prime for z = -14, -13,..., +10. Inx3-x2-17 replace re by x —10; we get a cubic which is a prime for re = 0, 1,. .., 24. "Annals'of Math., (2), 15, 1913, 63-5. wArchiv Math. Phys., (3), 25, 1916, 131-4. i°°Corresp. Math. Phys. (ed., Fuss), I, 1843, 595, letter to Euler, Nov. 18, 1752. 1MNovi Comm. Acad. Petrop., 9, 1762-3, 99; Comm. Arith., 1, 357. 102Mem. de Berlin, annee 1772, 36; Comm. Arith., 1, 584. 1030pera postuma, 1,1862, 185. In Pascal's Repertorium Hoheren Math., German transl. by Schepp, 1900,1, 518, it is stated incorrectly to be a prime for the first 17 values of x; likewise by Legendre, Throne dea nombres, 1798, 10; 1808, 11. 1MTheorie des nombres, 1798, 10, 304-312; ed. 2, 1808, 11, 279-285; ed. 3, 1830, I, 248-255; German transl. by Maser, I, 322-9. 104aNouv. Ann. Math., 3, 1844, 250. »*M&n. 1'Inst. Nat. Genevois, 5, 1857, No. 2, 7 pp. lMNouv. Corresp. Math., 5, 1879, 371; quoted in I'interm6diaire des math., 5, 1898, 114-5. 107L'interme<iiaire.des math., 6, 1899, 10-11. 108The same 40 primes as for x =0,..., 39, as noted by G. Lemaire, ibid.. 16 1909 p 197 "•/bid., 17, 1910, 271.