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A106856 Primes of the form x^2+xy+2y^2,with x and y nonnegative. 582
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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]. [T. D. Noe, Sep 01 2009]

REFERENCES

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

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

LINKS

Zak Seidov, Table of n,a(n) for n=1..1225

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

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)

Sequence in context: A158189 A218255 A085745 * A045387 A103255 A031385

Adjacent sequences:  A106853 A106854 A106855 * A106857 A106858 A106859

KEYWORD

nonn,easy

AUTHOR

T. D. Noe, May 09 2005, Apr 28 2008

EXTENSIONS

Removed old Mathematica programs T. D. Noe, Sep 09 2009

STATUS

approved

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Last modified April 18 06:39 EDT 2014. Contains 240706 sequences.