A040160 -id:A040160 - cp's OEIS frontend

A040159 Primes p such that x^5 = 2 has a solution mod p.

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 151, 157, 163, 167, 173, 179, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 277, 283, 293, 307, 313, 317, 337, 347, 349, 353

Offset: 1

N. J. A. Sloane

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

Cf. A001132, A040028, A040098, A040160.
Has same beginning as A042991 but is strictly different.
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^5 eq 2}]; // Bruno Berselli, Sep 12 2012

ok [p_]:=Reduce[Mod[x^5- 2, p]== 0, x, Integers]=!= False; Select[Prime[Range[180]], ok] (* Vincenzo Librandi, Sep 12 2012 *)

A083955 Numbers n > 1 such that n^5 - 2 has no prime factor > n.

Original entry on oeis.org

3557, 12038, 14810, 15424, 28456, 30742, 31540, 37665, 45602, 46883, 47879, 48152, 52196, 52617, 55265, 57902, 68306, 69032, 74925, 76262, 79562, 79984, 84569, 90442, 104867, 104956, 107213, 112570, 114614, 119477, 127634, 131072, 132466

Offset: 1

Klaus Brockhaus, May 09 2003

nonn

Also integers n > 1 for which there is no prime p > n such that x = n is a solution mod p of x^5 = 2, since the following equivalences hold for n > 1: There is a prime p > n such that n is a solution mod p of x^5 = 2 iff n^5 - 2 has a prime factor > n; n is a solution mod p of x^5 = 2 iff p is a prime factor of n^5 - 2 and p > n.

			12038 is a term since 12038^5 - 2 = 252796871460867395166 = 2*3*3*3*263*571*641*911*5849*9127 has no prime factor > 12038.

Cf. A040159, A040160, A065903.

t = {}; Do[If[Max[First/@FactorInteger[n^5-2]]Jayanta Basu, May 20 2013 *)
Select[Range[2,133000],Max[FactorInteger[#^5-2][[;;,1]]]<=#&] (* Harvey P. Dale, Mar 02 2023 *)

{for(n=2,133000,f=factor(n^5-2); if(f[matsize(f)[1],1]<=n,print1(n,",")))}

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).

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A040159 Primes p such that x^5 = 2 has a solution mod p.

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A083955 Numbers n > 1 such that n^5 - 2 has no prime factor > n.

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

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