1,1

Discriminant=-7. Binary quadratic forms ax^2+bxy+cy^2 have discriminant d=b^2-4ac. We consider sequences of primes produced by forms with -100<d<0, abs(b)<=a<=c and gcd(a,b,c)=1. When b is not zero, then there are two cases to consider: (1) nonnegative x and y and (2) x and y any integer. These restrictions yield 203 sequences of prime numbers, which are organized by discriminant below.

The Mathematica function QuadPrimes is useful for finding the primes less than "lim" represented by the quadratic form ax^2+bxy+cy^2 for any a, b, and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2+bxy+cy^2 <= lim. To find the primes generated by positive and negative x and y, merely compute the union of QuadPrimes[a,b,c,lim] and QuadPrimes[a,-b,c,lim]. [From T. D. Noe (noe(AT)sspectra.com), Sep 01 2009]

D. Cox, Primes of Form x^2 + n y^2, Wiley, 1989.

L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Contribution from T. D. Noe (noe(AT)sspectra.com), Sep 01 2009: (Start)

QuadPrimes[a_, b_, c_, lim_] := Module[{p, pLst={}, d, xMax, yMax}, d=b^2-4a*c; If[a>0 && c>0 && d<0, xMax=Ceiling[Sqrt[ -4*c*lim/d]]; yMax=Ceiling[Sqrt[ -4*a*lim/d]]; Do[p=a*x^2+b*x*y+c*y^2; If[p<=lim && PrimeQ[p], AppendTo[pLst, p]], {x, 0, xMax}, {y, 0, yMax}]]; Union[pLst]];

QuadPrimes[1, 1, 2, 1000] (End)

Cf. A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19),

A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881(d=-35), A040117, A068228, A106882 (d=-36),

A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55),

A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68),

A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256,

A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).

A139643, A139827 (the beginning of other collections of quadratic forms) [From T. D. Noe (noe(AT)sspectra.com), Sep 01 2009]

Sequence in context: A106927 A158189 A085745 this_sequence A045387 A103255 A031385

Adjacent sequences: A106853 A106854 A106855 this_sequence A106857 A106858 A106859

nonn,easy

T. D. Noe (noe(AT)sspectra.com), May 09 2005, Apr 28 2008

Removed old Mathematica programs T. D. Noe (noe(AT)sspectra.com), Sep 09 2009