A141375 -id:A141375 - cp's OEIS frontend

A107008 Primes of the form x^2 + 24*y^2.

73, 97, 193, 241, 313, 337, 409, 433, 457, 577, 601, 673, 769, 937, 1009, 1033, 1129, 1153, 1201, 1249, 1297, 1321, 1489, 1609, 1657, 1753, 1777, 1801, 1873, 1993, 2017, 2089, 2113, 2137, 2161, 2281, 2377, 2473, 2521, 2593, 2617, 2689, 2713

Offset: 1

T. D. Noe, May 09 2005

Presumably this is the same as primes congruent to 1 mod 24, so a(n) = 24*A111174(n) + 1. - N. J. A. Sloane, Jul 11 2008. Checked for all terms up to 2 million. - Vladimir Joseph Stephan Orlovsky, May 18 2011.
Discriminant = -96.
Also primes of the forms x^2 + 48*y^2 and x^2 + 72*y^2. See A140633. - T. D. Noe, May 19 2008
Primes of the quadratic form are a subset of the primes congruent to 1 (mod 24). [Proof. For 0 <= x, y <= 23, the only values mod 24 that x^2 + 24*y^2 can take are 0, 1, 4, 9, 12 or 16. All of these r except 1 have gcd(r, 24) > 1 so if x^2 + 24*y^2 is prime its remainder mod 24 must be 1.] - David A. Corneth, Jun 08 2020
More advanced mathematics seems to be needed to determine whether this sequence lists all primes congruent to 1 (mod 24). Note the significance of 24 being a convenient number, as described in A000926. See also Sloane et al., Binary Quadratic Forms and OEIS, which explains how the table in A139642 may be used for this determination. - Peter Munn, Jun 21 2020
Primes == 1 (mod 2^3*3) are the intersection of the primes == 1 (mod 2^3) in A007519 and the primes == 1 (mod 3) in A002476, by the Chinese remainder theorem. - R. J. Mathar, Jun 11 2020

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 143 terms from N. J. A. Sloane]
P. L. Clark, J. Hicks, H. Parshall, K. Thompson, GONI: primes represented by binary quadratic forms, INTEGERS 13 (2013) #A37
D. A. Cox, Primes of the form x^2 + n*y^2, A Wiley-Interscience publication, 1989
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. Voight, Quadratic forms that represent almost the same primes, Math. Comp. 76 (2007) 1589-1617

Subset of A033199 (2y here = y there).
Is this the same as A141375?
Cf. A000926, A002476, A007519, A111174, A139642.
See also the cross-references in A140633.

QuadPrimes[1, 0, 24, 10000] (* see A106856 *)

is(n) = isprime(n) && #qfbsolve(Qfb(1, 0, 24), n) == 2 \\ David A. Corneth, Jun 21 2020

Recomputed b-file, deleted incorrect Mma program. - N. J. A. Sloane, Jun 08 2014

A107003 Primes of the form 24n + 5.

Original entry on oeis.org

5, 29, 53, 101, 149, 173, 197, 269, 293, 317, 389, 461, 509, 557, 653, 677, 701, 773, 797, 821, 941, 1013, 1061, 1109, 1181, 1229, 1277, 1301, 1373, 1493, 1613, 1637, 1709, 1733, 1877, 1901, 1949, 1973, 1997, 2069, 2141, 2213, 2237, 2309, 2333, 2357, 2381, 2477

Offset: 1

T. D. Noe, May 09 2005

Primes of the form 5x^2+2xy+5y^2, with x and y any integer. Discriminant=-96. Also primes of the forms 5x^2+4xy+20y^2 and 5x^2+2xy+29y^2. See A140633. - T. D. Noe, May 19 2008

Also primes of the form -4*x^2+4*x*y+5*y^2, of discriminant -96 (as well as of the form 8*x^2+16*x*y+5*y^2). - Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

			29 is a member because we can write 29=-4*4^2+4*4*3+5*3^2 (or 29=8*1^2+16*1*1+5*1^2).

Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A141373, A141375, A141376 (d = -96).

Union[QuadPrimes2[5, 2, 5, 10000], QuadPrimes2[5, -2, 5, 10000]] (* see A106856 *)
Select[24*Range[0,200]+5,PrimeQ] (* Harvey P. Dale, Aug 25 2025 *)

select(n->n%24==5, primes(1000)) \\ Charles R Greathouse IV, Dec 07 2014

a(n) ~ 8n log n. - Charles R Greathouse IV, Dec 07 2014

Name and comment switched by Charles R Greathouse IV, Dec 07 2014

Edited by N. J. A. Sloane, Jul 14 2019

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

Original entry on oeis.org

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

Offset: 1

T. D. Noe, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - T. D. Noe, May 19 2008

			19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy] See Table II.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107132, A139827, A107003, A141375, A141376 (d=96).
See also A038872 (d=5),
A038873 (d=8),
A068228, A141123 (d=12),
A038883 (d=13),
A038889 (d=17),
A141158 (d=20),
A141159, A141160 (d=21),
A141170, A141171 (d=24),
A141172, A141173 (d=28),
A141174, A141175 (d=32),
A141176, A141177 (d=33),
A141178 (d=37),
A141179, A141180 (d=40),
A141181 (d=41),
A141182, A141183 (d=44),
A033212, A141785 (d=45),
A068228, A141187 (d=48),
A141188 (d=52),
A141189 (d=53),
A141190, A141191 (d=56),
A141192, A141193 (d=57),
A107152, A141302, A141303, A141304 (d=60),
A141215 (d=61),
A141111, A141112 (d=65),
A141336, A141337 (d=92),
A141338, A141339 (d=93),
A141161, A141163 (d=148),
A141165, A141166 (d=229),
A141167, A141167, A141167 (d=257).

[3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\16), if(isprime(t=w+16*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.

A141376 Primes of the form -x^2 + 8xy + 8y^2 (as well as of the form 15x^2 + 24xy + 8*y^2).

Original entry on oeis.org

23, 47, 71, 167, 191, 239, 263, 311, 359, 383, 431, 479, 503, 599, 647, 719, 743, 839, 863, 887, 911, 983, 1031, 1103, 1151, 1223, 1319, 1367, 1439, 1487, 1511, 1559, 1583, 1607, 1823, 1847, 1871, 2039, 2063, 2087, 2111, 2207, 2351, 2399, 2423, 2447, 2543

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

Discriminant = +96.
Values of the quadratic form are {0, 8, 12, 15, 20, 23} mod 24, so this is a subsequence of A134517. - R. J. Mathar, Jul 30 2008
Is this the same sequence as A134517?
Substituting 2y = y' gives the quadratic form A141171, so these terms are a subsequence of the terms in A141171. - R. J. Mathar, Jun 10 2020

			a(2)=47 because we can write 47 = -1^2 + 8*1*2 + 8*2^2 (or 47 = 15*1^2 + 24*1*1 + 8*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107003, A134517, A141171, A141373, A141375 (d = -96).

More terms from Arkadiusz Wesolowski, Jul 25 2012

cp's OEIS Frontend

A107008 Primes of the form x^2 + 24*y^2.

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A107003 Primes of the form 24n + 5.

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A141373 Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2).

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Magma

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A141376 Primes of the form -x^2 + 8*x*y + 8*y^2 (as well as of the form 15*x^2 + 24*x*y + 8*y^2).

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Author

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Examples

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Extensions

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

A141376 Primes of the form -x^2 + 8xy + 8y^2 (as well as of the form 15x^2 + 24xy + 8*y^2).