A014556 Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1.
2, 3, 5, 11, 17, 41
Offset: 1
References
- J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 225.
- J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
- I. N. Herstein and I. Kaplansky, Matters Mathematical, Chelsea, NY, 2nd. ed., 1978, see p. 38.
- F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983.
Links
- Aram Bingham, Ternary arithmetic, factorization, and the class number one problem, arXiv:2002.02059 [math.NT], 2020. See p. 8.
- Hung Viet Chu, Steven J. Miller, and Joshua M. Siktar, Composite Numbers in an Arithmetic Progression, arXiv:2411.03330 [math.HO], 2024. See p. 7.
- Brady Haran and Matt Parker, Caboose Numbers, Youtube video, June 2024.
- Harold M. Stark, A complete determination of the complex quadratic fields of class-number one, The Michigan Mathematical Journal 14.1 (1967): 1-27.
- Eric Weisstein's World of Mathematics, Lucky Number of Euler
- Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Programs
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Mathematica
A003173 = Union[Select[-NumberFieldDiscriminant[Sqrt[-#]] & /@ Range[200], NumberFieldClassNumber[Sqrt[-#]] == 1 &] /. {4 -> 1, 8 -> 2}]; a[n_] := (A003173[[n + 4]] + 1)/4; Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Jul 16 2012, after M. F. Hasler *) Select[Range[50],AllTrue[Table[m^2-m+#,{m,0,#-1}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 12 2017 *)
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PARI
is(n)=n>1 && qfbclassno(1-4*n)==1 \\ Charles R Greathouse IV, Jan 29 2013
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PARI
is(p)=for(n=1,p-1, if(!isprime(n*(n-1)+p),return(0))); 1 \\ naive; Charles R Greathouse IV, Aug 26 2022
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PARI
is(p)=for(n=1,sqrt(p/3)\/1, if(!isprime(n*(n-1)+p),return(0))); 1 \\ Charles R Greathouse IV, Aug 26 2022
Formula
a(n) = (A003173(n+3) + 1)/4. - M. F. Hasler, Nov 03 2008
Comments