420 History of the Theory of Numbers. [Chap, xvin 
R. D. CannichaeP for p*n — 1 (p an odd prime) and 2*-3m — 1. 
M. Bauer's^ paper was not available for report. 
Polynomials Representing Numerous Primes. 
Chr. Goldbach^^" noted that a polynomial fix) cannot represent primes 
exclusively, since the constant term would be unit}'-, whereas it is f{p) 
in /(x+p). 
L. Euler^"^ proved this by noting that, if /(«) =A,f(nA-\-a) is divisible 
by A. 
Euler^°- noted that x^ — t+41 is a prime for a: = 1, . . . , 40. 
Euler^"^ noted that a:- +2:+ 17 is a prime for x = 0, 1, . . ., 15 and [error] 
16; x'^+x+41 is a prime for x = 0, 1, . . ., 15. 
A. M. Legendre^^ noted that x^+a:+41 is a prime for x = 0, 1,. . ., 39, 
that 2x^+29 is a prime for x = 0, 1, . . ., 28, and gave a method of finding 
such functions. [Replacing x by x+1 in Euler's^°- function, we get 
X-+X+41.] If j8^-|-2(a+i3)x — 13x^ is a square only when x=0, and a and 
j3 are relatively prime, then a^+2aj3+14/3- is a prime or double a prime. 
He gave many such results. 
Chabert^"^" stated that 37i^-f 3n+l represents many primes for n small. 
G. 01tramare^°^ noted that x^+ax+6 has no prime divisor ^/z and 
hence is a prime when <fjr, if a^—4b is a quadratic non-residue of each of 
the primes 2, 3, . . ., ix. The function x^+ax+(a^+163)/4 is suitable to 
represent a series of primes. Taking x = 0, a=ii/v, he stated that u^+lQSv^ 
or its quotient by 4 gives more than 100 primes between 40 and 1763. 
H. LeLasseur^"^ verified that, for a prime A between 41 and 54000, 
x^+x+A does not represent primes exclusively for x = 0, 1,. . ., A— 2. 
E. B. Escott^"" noted that x"+x+41 gives primes not only forx = 0, 1, 
. . . , 39, but also^*^^ for x= — 1, — 2, . . . , —40. Hence, replacing x by x— 40, 
we get x^ — 79x + 1601, a prime for x = 0, 1, . . ., 79. Several such functions 
are given. 
Escott^"^ examined values of A much exceeding 54000 in x^+x+A 
without finding a suitable A>41. Legendre's^*^ first seven formulas for 
primes give composite numbers for a = 2, the eighth for a = 3, etc. Escott 
foundthatx^+x^+17isaprimeforx= — 14, — 13, . . ., +10. Inx^— x^ — 17 
replace x by x — 10; we get a cubic which is a prime for x = 0, 1, . . . , 24. 
"Annals of Math., (2), 15, 1913, 63-5. ^Archiv Math. Phvs., (3), 25, 1916, 131^. 
""Corresp. Math. Phys. (ed., Fuss), I, 1843, 595, letter to Euler, Nov. 18, 1752. 
'oiNovi Comm. Acad. Petrop., 9, 1762-3, 99; Comm. Arith., 1, 357. 
i^M^m. de Berlin, ann^e 1772, 36; Comm. Arith., 1, 584. 
"'K)pera postuma, I, 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; Uke- 
wise by Legendre, Th^orie des nombres, 1798, 10; i808, 11. 
iMTh^oric 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. '<»<'Nouv. Ann. Math., 3, 1844, 250. 
i^M^m. rinat. Nat. Genevois, 5, 1857, No. 2, 7 pp. 
•"Nouv. Corresp. Math., 5, 1879, 371; quoted in rinterm^diaire des math., 5, 1898, 114-5. 
'"L'intermddiaire des math., 6, 1899, 10-11. 
"•The same 40 primes as for x=0, . . ., 39, as noted by G. Lemaire, ibid., 16, 1909, p. 197. 
'"/Wd., 17, 1910, 271. 
