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7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303
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listen;
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internal format)
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OFFSET
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1,1
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COMMENTS
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Name was: Primes of the form x^2 + 3*x*y - 3*y^2 (as well as of the form x^2 + 5*x*y + y^2).
Discriminant = 21. Class number = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac and gcd(a,b,c)=1 (primitive).
Primes of the form 6n+1 which cannot be expressed as 7k-1, 7k-2, or 7k-4. a(n)^2 == 1 (mod 24). - Gary Detlefs, Jan 26 2014
Besides 7 (which divides 21), primes of the form p == 1 (mod 3) and either == 1 or 2 or 4 (mod 7). For the other class, the primes represented by the principal form [3, 3, -1] (or primitive forms equivalent to this) are besides 3 (which divides 21), congruent to 2 (mod 3) and also to 3, 5, 6 (mod 7). For the primes of both classes see A038893. - Wolfdieter Lang, Jun 19 2019
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REFERENCES
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Z. I. Borevich and I. R. Shafarevich, Number Theory.
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LINKS
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EXAMPLE
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a(1)=7 because we can write 7 = 2^2 + 3*2*1 - 3*1^2 (or 7 = 1^2 + 5*1*1 + 1^2).
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MAPLE
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f:=n->7*ceil((6*n+1)/7)-(6*n+1):for n from 1 to 220 do if isprime(6*n+1) and f(n)<>1 and f(n)<>2 and f(n)<>4 then print(6*n+1) fi od. # Gary Detlefs, Jan 26 2014
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MATHEMATICA
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xy[{x_, y_}]:={x^2+3x y-3y^2, y^2+3x y -3x^2}; Union[Select[Flatten[xy/@ Subsets[ Range[50], {2}]], #>0&&PrimeQ[#]&]] (* Harvey P. Dale, Feb 17 2013 *)
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PROG
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(Sage) # uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 3, -3])
Q.represented_positives(1326, 'prime') # Peter Luschny, Jun 24 2019
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CROSSREFS
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KEYWORD
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dead
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AUTHOR
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Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (laucabfer(AT)alum.us.es), Jun 12 2008
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EXTENSIONS
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STATUS
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approved
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