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A106856 Primes of the form x^2+xy+2y^2, with x and y nonnegative. 573
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)



Discriminant=-7. Binary quadratic forms ax^2+bxy+cy^2 have discriminant d=b^2-4ac.

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 positive definite 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, compute the union of QuadPrimes[a,b,c,lim] and QuadPrimes[a,-b,c,lim]. [T. D. Noe, Sep 01 2009]

For other programs see the "Binary Quadratic Forms and OEIS" link.


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.


N. J. A. Sloane and Zak Seidov, Table of n, a(n) for n = 1..10000 [The first 1225 terms were found by Zak Seidov]

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)


Do not use this program! 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 *)

Comments from N. J. A. Sloane, Jun 03 2014 (Start):

I believe this program QuadPrimes is wrong, since xMax should be sqrt(lmt/a)*(1+|b|/sqrt(-d)), as one sees by completing the square. The PARI program is also wrong for the same reason. This only makes a difference for primes close to the limit. Nevertheless, all the sequences listed below - especially the b-files - should be rechecked. It would be better to use the PARI library command qfbsolve, as in the second PARI program here.

For a specific example where QuadPrimes gives wrong answers, consider (in the PARI version) QuadPrimes(2, -1, 17, 100000).

This finds 858 primes, but because the range of x is too small, it misses the primes 99713 and 99833, as can be seen by running QuadPrimes(2, -1, 17, 100500). (End)

The following is a corrected version of the program. - N. J. A. Sloane, Jun 06 2014; corrected Jun 15 2014

QuadPrimes2[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]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; 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]];

QuadPrimes2[1, 1, 2, 1000]



# Lists ALL primes p represented by ax^2+bxy+cy^2 in the range 2 <= p <= prime(M) (where x and y may have any signs). The discriminant b^2-4ac should not be a square, but otherwise may be positive or negative. For square discriminants use the Maple code in A242660. - N. J. A. Sloane, Jun 03 2014

{ fc(a, b, c, M) = my(p, t1, t2, n); t1 = listcreate();

for(n=1, M, p = prime(n);

t2 = qfbsolve(Qfb(a, b, c), p); if(t2 == 0, , listput(t1, p)));



fc(1, 1, 2, 100); \\ Gives A045373 rather than A106856


Discriminants in the range -3 to -100: 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 (other collections of quadratic forms).

For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Cf. also A242660.

Sequence in context: A158189 A218255 A085745 * A045387 A103255 A031385

Adjacent sequences:  A106853 A106854 A106855 * A106857 A106858 A106859




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


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

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014



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Last modified July 31 09:51 EDT 2014. Contains 245083 sequences.