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A106856
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Primes of the form x^2+xy+2y^2,with x and y nonnegative.
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582
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2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821
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OFFSET
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1,1
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COMMENTS
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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, Sep 01 2009]
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REFERENCES
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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.
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LINKS
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Zak Seidov, Table of n,a(n) for n=1..1225
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MATHEMATICA
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QuadPrimes[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]]; Do[yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c); Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p] && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]]; QuadPrimes[1, 1, 2, 1000] (* a collaboration of T. D. Noe, Zak Seidov and Robert G. Wilson v *)
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PROG
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(PARI) QuadPrimes(a, b, c, lmt)={
my(p, d=b^2-4*a*c, lst=List());
for(x=0, sqrtint(lmt/a),
for(y=0, (sqrt(4*c*lmt + d*x^2)-b*x)/(c+c),
p = a*x^2 + b*x*y + c*y^2;
if(isprime(p), listput(lst, p))
)
);
vecsort(Vec(lst), , 8)
}; \\ Translation of Mathematica program
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CROSSREFS
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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, Sep 01 2009]
Sequence in context: A158189 A218255 A085745 * A045387 A103255 A031385
Adjacent sequences: A106853 A106854 A106855 * A106857 A106858 A106859
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KEYWORD
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nonn,easy
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AUTHOR
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T. D. Noe, May 09 2005, Apr 28 2008
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EXTENSIONS
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Removed old Mathematica programs T. D. Noe, Sep 09 2009
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STATUS
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approved
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