A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.
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
Offset: 1
A033212 Primes congruent to 1 or 19 (mod 30).
19, 31, 61, 79, 109, 139, 151, 181, 199, 211, 229, 241, 271, 331, 349, 379, 409, 421, 439, 499, 541, 571, 601, 619, 631, 661, 691, 709, 739, 751, 769, 811, 829, 859, 919, 991, 1009, 1021, 1039, 1051, 1069, 1129, 1171, 1201, 1231, 1249, 1279, 1291, 1321, 1381
Offset: 1
Comments
Theorem: Same as primes of the form x^2+15*y^2 (discriminant -60). Proof: Cox, Cor. 2.27, p. 36.
Equivalently, primes congruent to 1 or 4 (mod 15). Also x^2+xy+4y^2 is the principal form of (fundamental) discriminant -15. The only other class for -15 contains the form 2x^2+xy+2y^2 (A106859), in the other genus. - Rick L. Shepherd, Jul 25 2014
Three further theorems (these were originally stated as conjectures, but are now known to be theorems, thanks to the work of J. B. Tunnell - see link):
1. The same as primes of the form x^2-xy+4y^2 (discriminant -15) and x^2-xy+19y^2 (discriminant -75), both with x and y nonnegative. - T. D. Noe, Apr 29 2008
2. The same as primes of the form x^2+xy+19y^2 (discriminant -75), with x and y nonnegative. - T. D. Noe, Apr 29 2008
3. The same as primes of the form x^2+5xy-5y^2 (discriminant 45). - N. J. A. Sloane, Jun 01 2014
Also primes of the form x^2+7*x*y+y^2 (discriminant 45).
Lemma (Will Jagy, Jun 12 2014): If c is any (positive or negative) even number, then x^2 + x y + c y^2 and x^2 + (4 c - 1) y^2 represent the same odd numbers.
Proof: x (x + y) + c y^2 = odd, therefore x is odd, x + y odd, so y is even. Let y = 2 t. Then x( x + 2 t) + 4 c t^2 = x^2 + 2 x t + 4 c t^2 = (x+t)^2 + (4c-1) t^2 = odd. QED With c = 4, neither one represents 2, so x^2+15y^2 and x^2+xy+4y^2 represent the same primes.
Also, primes which are squares (mod 3*5). Subsequence of A191018. - David Broadhurst and M. F. Hasler, Jan 15 2016
References
- Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
- David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
Links
- Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
- J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
- D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
Crossrefs
Cf. A141785 (d=45), A033212 (Primes of form x^2+15*y^2), A038872(d=5), A038873 (d=8), A068228, A141123 (d=12), A038883 (d=13), A038889 (d=17), A141111, A141112 (d=65).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Programs
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Mathematica
QuadPrimes2[1, 0, 15, 10000] (* see A106856 *) Select[Prime@Range[250], MemberQ[{1, 19}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)
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PARI
select(n->n%30==1||n%30==19, primes(100)) \\ Charles R Greathouse IV, Nov 09 2012
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PARI
is(p)=issquare(Mod(p,15))&&isprime(p) \\ M. F. Hasler, Jan 15 2016
Formula
a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012
Extensions
Edited by N. J. A. Sloane, Jun 01 2014 and Oct 18 2014: added Tunnell document, revised entry, merged with A141184. The latter entry was submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008.
Typo in crossrefs fixed by Colin Barker, Apr 05 2015
A341786 Norms of prime ideals in Z[(1+sqrt(-15))/2], the ring of integers of Q(sqrt(-15)).
2, 3, 5, 17, 19, 23, 31, 47, 49, 53, 61, 79, 83, 107, 109, 113, 121, 137, 139, 151, 167, 169, 173, 181, 197, 199, 211, 227, 229, 233, 241, 257, 263, 271, 293, 317, 331, 347, 349, 353, 379, 383, 409, 421, 439, 443, 467, 499, 503, 541, 557, 563, 571, 587
Offset: 1
Comments
The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[(1+sqrt(-15))/2] has class number 2.
Consists of the primes congruent to 1, 2, 3, 4, 5, 8 modulo 15 and the squares of primes congruent to 7, 11, 13, 14 modulo 15.
For primes p == 1, 4 (mod 15), there are two distinct ideals with norm p in Z[(1+sqrt(-15))/2], namely (x + y*(1+sqrt(-15))/2) and (x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p; for p == 2, 8 (mod 15), there are also two distinct ideals with norm p, namely (p, x + y*(1+sqrt(-15))/2) and (p, x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p^2 with y != 0; (3, sqrt(-15)) and (5, sqrt(-15)) are respectively the unique ideal with norm 3 and 5; for p == 7, 11, 13, 14 (mod 15), (p) is the only ideal with norm p^2.
Examples
Let |I| be the norm of an ideal I, then: |(2, (1+sqrt(-15))/2)| = |(2, (1-sqrt(-15))/2)| = 2; |(3, sqrt(-15))| = 3; |(5, sqrt(-15))| = 5; |(17, 7+4*sqrt(-15))| = |(17, 7-4*sqrt(-15))| = 17; |(2 + sqrt(-15))| = |(2 - sqrt(-15))| = 19; |(23, 17+4*sqrt(-15))| = |(23, 17-4*sqrt(-15))| = 23; |(4 + sqrt(-15))| = |(4 - sqrt(-15))| = 31.
Links
- Jianing Song, Table of n, a(n) for n = 1..10000
Crossrefs
The number of distinct ideals with norm n is given by A035175.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), this sequence (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.
Programs
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PARI
isA341786(n) = my(disc=-15); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
A028955 Numbers represented by quadratic form with Gram matrix [ 4, 1; 1, 4 ] (divided by 2).
0, 2, 3, 5, 8, 12, 17, 18, 20, 23, 27, 30, 32, 38, 45, 47, 48, 50, 53, 57, 62, 68, 72, 75, 80, 83, 92, 93, 95, 98, 102, 107, 108, 113, 120, 122, 125, 128, 137, 138, 147, 152, 153, 155, 158, 162, 167, 170, 173, 180, 183, 188, 192, 197, 200, 207, 212, 218, 227, 228
Offset: 1
Comments
Numbers of the form 2*x^2 + x*y + 2*y^2, of discriminant -15. - N. J. A. Sloane, Jun 01 2014
8*a(n) is of the form z^2 + 15*y^2, where z = 4*x + y. [Bruno Berselli, Jul 12 2014]
Examples
32 is in the sequence because it can be written in the form 2*2^2+2*3+2*3^2, and hence 8*32 = 11^2+15*3^2.
Links
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Formula
a(x, y) = (4x^2 + 2xy + 4y^2)/2; x, y any integer.
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000
A106858 Primes of the form 2x^2+xy+2y^2 with x and y nonnegative.
2, 5, 23, 83, 107, 137, 173, 257, 293, 347, 353, 467, 503, 617, 647, 653, 743, 797, 857, 953, 983, 1223, 1277, 1283, 1307, 1427, 1487, 1493, 1523, 1553, 1637, 1787, 1877, 1913, 1997, 2003, 2027, 2213, 2237, 2243, 2393, 2423, 2447, 2657, 2663
Offset: 1
Comments
Discriminant=-15.
Links
- Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 1000 terms from N. J. A. Sloane]
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Programs
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Mathematica
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]]; t2 = QuadPrimes2[2, 1, 2, 350000]; Length[t2] t2[[Length[t2]]] For[n=1, n <= 2000, n++, Print[n, " ", t2[[n]]]] (* From N. J. A. Sloane, Jun 17 2014 *)
Extensions
Replace Mma program by a correct program, recomputed and extended b-file. - N. J. A. Sloane, Jun 17 2014
A020678 Numbers of form 3 x^2 + 5 y^2.
0, 3, 5, 8, 12, 17, 20, 23, 27, 32, 45, 47, 48, 53, 57, 68, 72, 75, 80, 83, 92, 93, 95, 107, 108, 113, 120, 125, 128, 137, 147, 152, 153, 155, 167, 173, 180, 183, 188, 192, 197, 200, 207, 212, 227, 228, 233, 237, 243, 245, 248, 255, 257, 263, 272, 288, 293, 300, 305, 317
Offset: 1
Comments
Discriminant -60.
Links
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Crossrefs
The primes in this sequence are given by A106859 (excluding 2). - N. J. A. Sloane, Jun 01 2014
Programs
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Mathematica
With[{upto=320},Select[Union[3#[[1]]^2+5#[[2]]^2&/@ Tuples[ Range[ 0, Ceiling[ Sqrt[upto]]],2]],#<+upto&]] (* Harvey P. Dale, May 22 2015 *)
A343241 Primes congruent to 2 or 8 modulo 15.
2, 17, 23, 47, 53, 83, 107, 113, 137, 167, 173, 197, 227, 233, 257, 263, 293, 317, 347, 353, 383, 443, 467, 503, 557, 563, 587, 593, 617, 647, 653, 677, 683, 743, 773, 797, 827, 857, 863, 887, 947, 953, 977, 983
Offset: 1
Comments
This sequence is the complement of A033212 (primes congruent to 1 or 4 mod(15)) relative to the primes p with Jacobi(p|15) = +1 (A191018).
There is neither a solution x of the congruence x^2 == a(n) (mod 3) nor of x^2 == a(n) (mod 5) (the Legendre symbols are -1 in both cases, and Jacobi(a(n)|15) = +1).
Programs
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Mathematica
Select[Range[1000], PrimeQ[#] && MemberQ[{2, 8}, Mod[#, 15]] &] (* Amiram Eldar, May 20 2021 *)
Comments
References
Links
Crossrefs
Programs
Mathematica
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] && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]]; QuadPrimes2[1, 1, 2, 1000] (This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014) max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)PARI
Extensions