A042967 -id:A042967 - cp's OEIS frontend

A059667 Primes p such that x^49 = 2 has no solution mod p, but x^7 = 2 has a solution mod p.

4999, 6959, 7351, 11467, 15583, 16073, 20483, 21169, 21757, 30773, 35771, 37339, 38711, 41161, 45179, 46649, 48119, 51157, 51647, 57527, 58997, 64877, 75167, 75853, 80263, 83791, 84869, 85751, 86927, 93983, 95747, 105253, 110251, 115837

Offset: 1

Klaus Brockhaus, Feb 04 2001

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..76

Cf. A042966, A042967, A070179 - A070188.

Select[Prime[Range[PrimePi[120000]]], ! MemberQ[PowerMod[Range[#], 49, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 7, #], Mod[2, #]] &] (* Vincenzo Librandi, Sep 21 2013 *)

PARI
```
forprime(p=2,116000,x=0; while(x
				
```

N=10^6;  default(primelimit,N);
ok(p, r, k1, k2)={
    if (  Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
    if (  Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
    return(1);
}
forprime(p=2,N, if (ok(p,2,7,7^2),print1(p,", ")));
\\ Joerg Arndt, Sep 21 2012

A042966 Primes p such that x^7 = 2 has a solution mod p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 199, 223, 227, 229, 233, 241, 251, 257, 263, 269, 271, 277, 283, 293, 307, 311, 313, 317, 331

Offset: 1

N. J. A. Sloane

Coincides with sequence of "primes p such that x^49 = 2 has a solution mod p" for first 572 terms, then diverges.
Complement of A042967 relative to A000040. - Vincenzo Librandi, Sep 13 2012
a(98) = 631 is the first such prime that is congruent to 1 (mod 7). - Georg Fischer, Jan 06 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for related sequences

Cf. A000040, A042967.

For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

[p: p in PrimesUpTo(400) | exists{x: x in ResidueClassRing(p) | x^7 eq 2}]; // Bruno Berselli, Sep 12 2012

ok[p_]:= Reduce[Mod[x^7 - 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

A190616 Number of normal bases in GF(2^n) that are Gaussian normal bases.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 0, 3, 8, 3, 16, 5, 16, 15, 0, 17, 48, 27, 128, 63, 192, 89, 0, 205, 637, 171, 1011, 565

Offset: 1

Joerg Arndt, May 14 2011

A type-t Gaussian normal basis (GNB) exists for GF(2^n) if p=n*t+1 is prime and gcd(n,(p-1)/ord(2 mod p))==1.  In practice one finds (for fixed n) infinitely many t corresponding to some GNB.  As there are only finitely many normal bases for fixed n the GNBs for different t are not in general different but correspond to a finite set of field polynomials.  This sequence gives the number of field polynomials (equivalently, mod-2 reduced multiplication matrices) that correspond to some GNB.
The sequence was computed by determining all field polynomials for types t <= n*500 and discarding duplicate polynomials.  Note that there is no guarantee that the used bound (500*n) leads to discovery of all polynomials.
An efficient method to determine (for fixed n) whether two types, say t1 and t2, correspond to the same polynomial would be of great interest.
A computation using the bound t<=2000 gave a(22)=192 (the old value was 191), so the sequence was corrected past that term and truncated after a(29). [Joerg Arndt, May 16 2011]

			For n=5 there is just one field polynomial (x^5 + x^4 + x^2 + x + 1),
  for p in {11, 31, 41, 61, 71, 101, 131, ...} (A040160).
For n=7 there is just one field polynomial (x^7 + x^6 + x^4 + x + 1),
  for p in {29, 43, 71, 113, 127, 197,...} (A042967).
For n=11 there are three GNBs:
x^11 + x^10 + x^8 + x^4 + x^3 + x^2 + 1
  for p in {23, 463, 661, 859, 881, 1409, 1453, 2179, ...},
x^11 + x^10 + x^8 + x^5 + x^2 + x + 1
  for p in {67, 89, 353, 727, 947, 1277, 1607, 1783, 1871, ...}, and
x^11 + x^10 + x^8 + x^7 + x^6 + x^5 + 1
  for p in {199, 397, 419, 617, 683, 991, 1123, 2003, 2069, 2113, ...}.

Joerg Arndt, Fxtbook, section 42.9 "Gaussian normal bases", pp. 914-920.

a(8*n) = 0 (there is no GNB for multiples of eight).

cp's OEIS Frontend

A059667 Primes p such that x^49 = 2 has no solution mod p, but x^7 = 2 has a solution mod p.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A042966 Primes p such that x^7 = 2 has a solution mod p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A190616 Number of normal bases in GF(2^n) that are Gaussian normal bases.

Views

Author

Keywords

Comments

Examples

Links

Formula