A070184 -id:A070184 - cp's OEIS frontend

A045316 Primes p such that x^8 = 2 has no solution mod p.

3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389

Offset: 1

N. J. A. Sloane

Complement of A045315 relative to A000040. Coincides for the first 140 terms with the sequence of primes p such that x^16 = 2 has no solution mod p (first divergence is at 1217, cf. A059287). - Klaus Brockhaus, Jan 26 2001

Differs from A059349 (x^32 == 2 (mod p) has no solution) first at a(37) = A059349(38), the term A059349(37) = 257 which is not in this sequence. See A070184 for all such terms. - M. F. Hasler, Jun 21 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A045315 (complement in the primes), A059287.

Subsequence of A059349 (same with x^32), complement is A070184.

[p: p in PrimesUpTo(500) | not exists{x : x in ResidueClassRing(p) | x^8 eq 2} ]; // Vincenzo Librandi, Sep 19 2012

ok[p_]:= Reduce[Mod[x^8 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 19 2012 *)

select( {is_A045316(p)=Mod(2,p)^(p\gcd(8,p-1))!=1 && p>2}, primes(199)) \\ Append "&& isprime(p)" if that's not known. - M. F. Hasler, Jun 22 2024

A059349 Primes p such that x^32 = 2 has no solution mod p.

Original entry on oeis.org

3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389, 397, 401, 409, 419

Offset: 1

Klaus Brockhaus, Jan 27 2001

Complement of A049564 relative to A000040.
Differs from A014662 first at p=6529, then at p=21569. [R. J. Mathar, Oct 05 2008]
Differs from A045316 (x^8 == 2 (mod p) has no solution) first at a(37) = 257 which is not a term of A045316. See A070184 for all such terms. - M. F. Hasler, Jun 21 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A049564, A216747.

Cf. A070184 = (this sequence) \ A045316.

[p: p in PrimesUpTo(450) | not exists{x : x in ResidueClassRing(p) | x^32 eq 2 }]; // Vincenzo Librandi, Sep 20 2012

ok[p_] := Reduce[Mod[x^32 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[100]], ok ] (* Vincenzo Librandi, Sep 20 2012  *)

A070185 Primes p such that x^9 = 2 has a solution mod p, but x^(9^2) = 2 has no solution mod p.

Original entry on oeis.org

3943, 5347, 11287, 12853, 14149, 17659, 20143, 21061, 21277, 23059, 23599, 25759, 26407, 26731, 29863, 32833, 33751, 35803, 37747, 38287, 39367, 39799, 46441, 47737, 47791, 54919, 57781, 59887, 61291, 62047, 63127, 65557, 68311, 71443, 73063, 75169, 78301, 79273, 82351, 84457, 84673, 86077, 88129, 90289

Offset: 1

Klaus Brockhaus, Apr 29 2002

nonn

Cf. A049596, A059667, A070179 - A070184, A070186 - A070188.

Cf. A059354.

PARI
```
forprime(p=2,72000,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,2,9,9^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

A059287 Primes p such that x^16 = 2 has no solution mod p, but x^8 = 2 has a solution mod p.

Original entry on oeis.org

1217, 1249, 1553, 1777, 2833, 4049, 4273, 4481, 4993, 5297, 6449, 6481, 6689, 7121, 8081, 8609, 9137, 9281, 9649, 10337, 10369, 10433, 11329, 11617, 11633, 12241, 12577, 13121, 13441, 13633, 14321, 14753, 15121, 15569, 16417, 16433, 16673

Offset: 1

Klaus Brockhaus, Jan 25 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..150

Cf. A000040, A045315, A045316.

Cf. A070184 (same with x^64 instead of x^16).

[p: p in PrimesUpTo(17000) | not exists{x: x in ResidueClassRing(p) | x^16 eq 2} and exists{x: x in ResidueClassRing(p) | x^8 eq 2}]; // Vincenzo Librandi, Sep 21 2012

Select[Prime[Range[PrimePi[20000]]], !MemberQ[PowerMod[Range[#], 16, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 8, #], Mod[2, #]]&] (* Vincenzo Librandi, Sep 21 2013 *)

select( {is_A059287(p)=Mod(2,p)^(p\gcd(8,p-1))==1&&Mod(2,p)^(p\gcd(16,p-1))!=1}, primes(1999)) \\ Could any composite number pass this test? - M. F. Hasler, Jun 22 2024

from itertools import islice
from sympy import is_nthpow_residue, nextprime
def A059287_gen(startvalue=2): # generator of terms >= startvalue
    p = max(1,startvalue-1)
    while (p:=nextprime(p)):
        if is_nthpow_residue(2,8,p) and not is_nthpow_residue(2,16,p):
            yield p
A059287_list = list(islice(A059287_gen(),10)) # Chai Wah Wu, Jun 23 2024

A070183 Primes p such that x^6 = 2 has a solution mod p, but x^(6^2) = 2 has no solution mod p.

Original entry on oeis.org

17, 41, 137, 401, 433, 449, 457, 521, 569, 641, 761, 809, 857, 919, 929, 953, 977, 1361, 1409, 1423, 1657, 1697, 1999, 2017, 2081, 2143, 2153, 2287, 2297, 2417, 2609, 2633, 2729, 2753, 2777, 2791, 2801, 2897, 2953, 3041, 3209, 3329, 3457, 3593, 3617

Offset: 1

Klaus Brockhaus, Apr 29 2002

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040992, A049568, A059264, A059667, A070179, A070180, A070181, A070182, A070184, A070185, A070186, A070187, A070188.

[p: p in PrimesUpTo(5000) | not exists{x: x in ResidueClassRing(p) | x^36 eq 2} and exists{x: x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 21 2012

select(p -> isprime(p) and [msolve(x^6=2,p)]<>[] and [msolve(x^36=2,p)]=[] , [seq(i,i=3..10^4,2)]); # Robert Israel, May 13 2018

PARI
```
forprime(p=2,3700,x=0; while(x
				
```

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=2,10^4, if (ok(p,2,6,6^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

from itertools import count, islice
from sympy import nextprime, is_nthpow_residue
def A070183_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1),2)
    while True:
        if is_nthpow_residue(2,6,p) and not is_nthpow_residue(2,36,p):
            yield p
        p = nextprime(p)
A070183_list = list(islice(A070183_gen(),20)) # Chai Wah Wu, May 02 2024

A252279 Primes p congruent to 1 mod 16 such that x^8 = 2 has a solution mod p.

Original entry on oeis.org

257, 337, 881, 1217, 1249, 1553, 1777, 2113, 2593, 2657, 2833, 4049, 4177, 4273, 4481, 4513, 4721, 4993, 5297, 6353, 6449, 6481, 6529, 6689, 7121, 7489, 8081, 8609, 9137, 9281, 9649, 10177, 10337, 10369, 10433, 10657, 11329, 11617, 11633, 12049, 12241, 12577

Offset: 1

Arkadiusz Wesolowski, Dec 16 2014

nonn

For a prime p congruent to 1 mod 16, the number 2 is an octavic residue mod p if and only if there are integers x and y such that x^2 + 256*y^2 = p.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Index entries for related sequences

Subsequence of A045315.

Has A070184 as a subsequence.

[p: p in PrimesUpTo(12577) | p mod 16 eq 1 and exists(t){x : x in ResidueClassRing(p) | x^8 eq 2}]; // Arkadiusz Wesolowski, Dec 19 2020

isok(p) = isprime(p) && (Mod(p, 16) == 1) && ispower(Mod(2, p), 8); \\ Michel Marcus, Dec 19 2020

A373468 Primes such that x^16 = 2 has a solution in Z/pZ, but x^32 = 2 does not.

Original entry on oeis.org

257, 2113, 2657, 7489, 10177, 15073, 18593, 23041, 25121, 25409, 25537, 25793, 27809, 30881, 30977, 32321, 37409, 38273, 41729, 43649, 51137, 51361, 54721, 59809, 63841, 67073, 67489, 75553, 77569, 83009, 86561, 92641, 94049, 94433, 95713, 101281, 102241

Offset: 1

M. F. Hasler, Jun 22 2024

nonn

			For p = 257, the equation x^16 = 2 has solutions 27, 41, 54, ... in Z/pZ, but x^32 can only be 0, +-1, +-4, +-16, +-64 (mod p).

Cf. A059287 (similar for x^8 vs x^16).

Subsequence of A070184 which is a subsequence of A252279.

select( {is_A373468(p)=Mod(2,p)^(p\gcd(16,p-1))==1&&Mod(2,p)^(p\gcd(32,p-1))!=1}, primes(19999))

from itertools import islice
from sympy import nextprime, is_nthpow_residue
def A373468_gen(startvalue=2): # generator of terms >= startvalue
    p = max(1,startvalue-1)
    while (p:=nextprime(p)):
        if is_nthpow_residue(2,16,p) and not is_nthpow_residue(2,32,p):
            yield p
A373468_list = list(islice(A373468_gen(),10)) # Chai Wah Wu, Jun 23 2024

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A045316 Primes p such that x^8 = 2 has no solution mod p.

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A059349 Primes p such that x^32 = 2 has no solution mod p.

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A070185 Primes p such that x^9 = 2 has a solution mod p, but x^(9^2) = 2 has no solution mod p.

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A059287 Primes p such that x^16 = 2 has no solution mod p, but x^8 = 2 has a solution mod p.

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A070183 Primes p such that x^6 = 2 has a solution mod p, but x^(6^2) = 2 has no solution mod p.

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A252279 Primes p congruent to 1 mod 16 such that x^8 = 2 has a solution mod p.

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A373468 Primes such that x^16 = 2 has a solution in Z/pZ, but x^32 = 2 does not.

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