Here are some excerpts from A17 of UPINT3. Comments welcome. R. --------------------------- We have already mentioned (in {\bf A1}) Euler's famous formula $n^2+n+41$. In some sense this is best possible, but quadratic expressions with positive discriminant can yield even longer sequences of prime values (though some of them may be negative). Gilbert Fung gives $47n^2-1701n+10181$, $0\leq n\leq 42$, $\Delta = 979373$ and Russell Ruby $36n^2-810n+2753$, $0\leq n\leq 44$, $\Delta=2^23^27213$. The first 1000 values of Euler's formula include 581 primes. Edgar Karst beats this with 598 values of $2n^2-199$ and in a 91-01-01 letter, Stephen Williams announces 602 prime values of $2n^2-1000n-2609$. The corresponding numbers among the first 10000 values are 4148, 4373 and 4151. However, what is significant is not the actual density over the first so many values, which clearly has to tend to zero in all cases, but the {\bf asymptotic} density, which, if we believe Hardy \& Littlewood (see {\bf A1}), is always $c\sqrt n/\ln n$, and the best that can be done \hGidx{asymptotic density} is to make the value of $c$ as large as possible. Shanks has calculated $c=3.3197732$ for Euler's formula and $c=3.6319998$ for a polynomial $x^2+x+27941$ found by Beeger. Fung \& Williams (see reference at {\bf A1} and the references they give) have achieved $c=5.0870883$ with the formula $x^2+x+132874279528931$. Nigel Boston \& Marshall L.~Greenwood, Quadratics representing primes, {\it Amer.\ Math.\ Monthly}, {\bf102}(1995) 595--599; {\it MR} {\bf96g}:11154. Gilbert W.~Fung \& Hugh Cowie Williams, Quadratic polynomials which have a high density of prime values, {\it Math.\ Comput.}, {\bf 55}(1990) 345--353; {\it MR} {\bf 90j}:11090. Betty Garrison, Polynomials with large numbers of prime values, {\it Amer.\ Math.\ Monthly}, {\bf 97}(1990) 316--317; {\it MR} {\bf 91i}:11124. P.~Goetgheluck, On cubic polynomials giving many primes, {\it Elem.\ Math.}, {\bf 44}(1989) 70--73; {\it MR} {\bf 90j}:11014. Masaki Kobayashi, Prime producing quadratic polynomials and class-number one problem for real quadratic fields, {\it Proc.\ Japan Acad.\ Ser.\ A Math.\ Sci.}, {\bf 66}(1990) 119--121; {\it MR} {\bf 91i}:11140. D.~H.~Lehmer, On the function $x^2+x+A$, {\it Sphinx}, {\bf 6}(1936) 212--214; {\bf 7}(1937) 40. S.~Louboutin, R.~A.~Mollin \& H.~C.~Williams, Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing polynomials and quadratic residue covers, {\it Canad.\ J.\ Math.}, {\bf44}(1992) 1--19. Richard A.~Mollin, Prime-producing quadratics, {\it Amer.\ Math.\ Monthly}, {\bf104}(1997) 529--544; {\it MR} {\bf98h}:11113. Richard A.~Mollin \& Hugh Cowie Williams, Quadratic nonresidues and prime-producing polynomials, {\it Canad.\ Math.\ Bull.}, {\bf 32}(1989) 474--478; {\it MR} {\bf 91a}:11009. [see also {\it Number Theory}, de Gruyter, 1989, 654--663 and {\it Nagoya Math.\ J.}, {\bf112}(1988) 143--151.] Joe L.~Mott \& Kermit Rose, Prime-producing cubic polynomials, {\it Ideal theoretic methods in commutative algebra $($Columbia MO} 1999) 281--317, {\it Lecture Notes in Pure and Appl.\ Math.}, {\bf220} Dekker, New York, 2001; {\it MR} {\bf2002c}:11140. ----------------------------- On Wed, 26 May 2004, M. Stay wrote:
http://www.primepuzzles.net/problems/prob_012.htm mentions the record-holding prime-producing polynomials for consecutive X
f(X) = 36 X^2 - 810 X + 2753 for which |f(X)|, 0 <= X <= 44 is prime (36X^2-2358X+36809 produces the same primes in reverse order!)
and f(X) = 47 X^2 - 1701 X + 10181 for which f(X) is prime for 0 <= X <= 42 -- Mike Stay staym@clear.net.nz http://www.cs.auckland.ac.nz/~msta039
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