A086965 -id:A086965 - cp's OEIS frontend

A106867 Primes of the form 2x^2 + xy + 3*y^2.

2, 3, 13, 29, 31, 41, 47, 71, 73, 127, 131, 139, 151, 163, 179, 193, 197, 233, 239, 257, 269, 277, 311, 331, 349, 353, 397, 409, 439, 443, 461, 487, 491, 499, 509, 541, 547, 577, 587, 601, 647, 653, 673, 683, 739, 761, 811, 823, 857, 859, 863, 887, 929, 947

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -23.
Primes p such that the polynomial x^3-x-1 is irreducible over Zp. The polynomial discriminant is also -23. - T. D. Noe, May 13 2005
Also, primes p such that tau(p) = A000594(p) == -1 (mod 23). [A proof can probably be found in van der Blij (1952). Thanks to Juan Arias-de-Reyna for this reference. - N. J. A. Sloane, Nov 29 2016]

F. van der Blij, Binary quadratic forms of discriminant -23. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14, (1952). 498-503; Math. Rev. MR0052462.
John Raymond Wilton, "Congruence properties of Ramanujan's function τ(n)." Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. R. Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only.

Cf. A086965 (number of distinct zeros of x^3-x-1 mod prime(n)).
Cf. also A000594.
These are the primes in A028929.

Union[QuadPrimes2[2, 1, 3, 10000], QuadPrimes2[2, -1, 3, 10000]] (* see A106856 *)

forprime(p=2,10^4,if(0==#polrootsmod(x^3-x-1,p),print1(p,", "))); /* Joerg Arndt, Jul 27 2011 */

forprime(p=2,10^4,if(polisirreducible(Mod(1, p)*(x^3-x-1)), print1(p, ", ") ) ); /* Joerg Arndt, Mar 30 2013 */

from itertools import count, islice
from sympy import prime, GF, Poly
from sympy.abc import x
def A106867_gen(): # generator of terms
    return filter(lambda p:Poly(x**3-x-1,domain=GF(p)).is_irreducible, (prime(i) for i in count(1)))
A106867_list = list(islice(A106867_gen(),20)) # Chai Wah Wu, Nov 11 2022

A086937 Number of distinct zeros of x^2-x-1 mod prime(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 2

Offset: 1

N. J. A. Sloane, Sep 23 2003

nonn

For the prime modulus 5, the polynomial can be factored as (x+2)^2, showing that x=3 is a zero of multiplicity 2. The discriminant of the polynomial is 5. Also note how this sequence is related to the Fibonacci sequence A051830; for n>3, a(n) = 2*A051830(n). - T. D. Noe, Aug 13 2004

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.

Cf. A086965, A086966, A086967.

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 105}] (* T. D. Noe, Sep 24 2003 *)

If p = prime(n), a(n) = A080891(p) + 1.

Corrected and extended by T. D. Noe, Sep 24 2003

A086966 Number of distinct zeros of x^4-x-1 mod prime(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 0, 2, 1, 0, 0, 2, 1, 1, 2, 0, 0, 2, 4, 1, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 2, 2, 2, 1, 1, 0, 0, 2, 1, 2, 2, 1, 1, 1, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 2, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 0, 2, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 2, 1, 1, 0

Offset: 1

N. J. A. Sloane, Sep 24 2003

nonn

For the prime modulus 283, the polynomial can be factored as (x+18) (x+168) (x+190)^2, showing that x=93 is a zero of multiplicity 2. The discriminant of the polynomial is 283. - T. D. Noe, Aug 12 2004

Robert Israel, Table of n, a(n) for n = 1..10000
J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see pp. 433-434.

Cf. A086937, A086965, A086967.

f:= n -> nops([msolve(x^4-x-1,ithprime(n))]):
map(f, [$1..100]); # Robert Israel, Aug 10 2023

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^4-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 105}] (* T. D. Noe, Sep 24 2003 *)

More terms from T. D. Noe, Sep 24 2003

A086967 Number of distinct zeros of x^5-x-1 mod prime(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 1, 1, 0, 0, 0, 2, 1, 3, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 3, 2, 0, 1, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 3, 2, 1, 2, 1, 0, 2, 2, 0, 1, 0, 2, 0, 0, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 3, 1, 3, 1, 3, 0, 0, 1, 0, 1

Offset: 1

N. J. A. Sloane, Sep 24 2003

nonn

For the prime modulus 19, the polynomial can be factored as (x+6)^2 (x^3+7x^2+13x+10), showing that x=13 is a zero of multiplicity 2. For the prime modulus 151, the polynomial can be factored as (x+9) (x+39)^2 (x^2+64x+61), showing that x=112 is a zero of multiplicity 2. The discriminant of the polynomial is 2869=19*151. - T. D. Noe, Aug 12 2004

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 435.

Cf. A086937, A086965, A086966.

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^5-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 100}] (from T. D. Noe)

More terms from T. D. Noe, Sep 24 2003

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A106867 Primes of the form 2*x^2 + x*y + 3*y^2.

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A086937 Number of distinct zeros of x^2-x-1 mod prime(n).

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A086966 Number of distinct zeros of x^4-x-1 mod prime(n).

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A086967 Number of distinct zeros of x^5-x-1 mod prime(n).

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A106867 Primes of the form 2x^2 + xy + 3*y^2.