A040098 -id:A040098 - cp's OEIS frontend

A045315 Primes p such that x^8 = 2 has a solution mod p.

2, 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 257, 263, 271, 311, 337, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 601, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919, 937, 967, 983, 991, 1031, 1039

Offset: 1

N. J. A. Sloane

Coincides with the sequence of "primes p such that x^16 = 2 has a solution mod p" for first 58 terms (and then diverges).

Complement of A045316 relative to A000040. - Vincenzo Librandi, Sep 13 2012

A. Aigner, Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und -2, Deutsche Math. 4 (1939), 44-52; FdM 65 - I (1939), 112.

T. D. Noe, Table of n, a(n) for n = 1..1000
H. Hasse, Der 2^n-te Potenzcharakter von 2 im Koerper der 2^n-ten Einheitswurzeln, Rend. Circ. Matem. Palermo (2), 7 (1958), 185-243.
Franz Lemmermeyer, Bibliography on Reciprocity Laws
A. L. Whiteman, The sixteenth power residue character of 2, Canad. J. Math. 6 (1954), 364-373; Zbl 55.27102.
Index entries for related sequences

Cf. A000040, A001132, A040028, A040098, A045316.

[p: p in PrimesUpTo(1100) | exists(t){x : x in ResidueClassRing(p) | x^8 eq 2}]; // Vincenzo Librandi, Sep 13 2012

ok[p_] := Reduce[ Mod[x^8-2, p] == 0, x, Integers] =!= False; Select[ Prime[ Range[200] ], ok] (* Jean-François Alcover, Nov 28 2011 *)

is(n)=isprime(n) && ispower(Mod(2,n),8) \\ Charles R Greathouse IV, Feb 08 2017

A072936 Primes p that do not divide 2^x+1 for any x>=1.

Original entry on oeis.org

2, 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 263, 271, 311, 337, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 601, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919, 937, 967, 983, 991, 1031, 1039, 1063

Offset: 1

Benoit Cloitre, Aug 20 2002

nonn

Also, primes p such that p^2 does not divide 2^x+1 for any x>=1.
A prime p cannot divide 2^x+1 if the multiplicative order of 2 (mod p) is odd. - T. D. Noe, Aug 22 2004
Differs from A049564 first at p=6529, which is the 250th entry in A049564 related to 279^32 =2 mod 6529, but absent here because 6529 divides 2^51+1. [From R. J. Mathar, Sep 25 2008]

A. K. Devaraj, "Euler's Generalization of Fermat's Theorem-A Further Generalization", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A040098, A049096, A014664 (multiplicative order of 2 mod n-th prime).

Edited by T. D. Noe, Aug 22 2004

A065902 Smallest prime p such that n is a solution mod p of x^4 = 2, or 0 if no such prime exists.

Original entry on oeis.org

7, 79, 127, 7, 647, 2399, 23, 937, 4999, 14639, 1481, 28559, 19207, 23, 31, 47, 73, 18617, 79999, 194479, 117127, 5711, 165887, 73, 4663, 113, 233, 707279, 47, 40153, 524287, 191, 167, 257, 439, 267737, 45329, 2313439, 182857, 2825759, 1555847

Offset: 2

Klaus Brockhaus, Nov 28 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for n > 1: There is a prime p such that n is a solution mod p of x^4 = 2 iff n^4 - 2 has a prime factor > n; n is a solution mod p of x^4 = 2 iff p is a prime factor of n^ 4 - 2 and p > n. n^4 - 2 has at most three prime factors > n, so these factors are the only primes p such that n is a solution mod p of x^4 = 2. The first zero is at n = 1689 (cf. A065903 ). For n such that n^4 - 2 has one resp. two resp. three prime factors > n; cf. A065904 resp. A065905 resp. A065906.

			a(16) = 31, since 16 is a solution mod 31 of x^4 = 2 and 16 is not a solution mod p of x^4 = 2 for primes p < 31. Although 16^4 = 2 (mod 7), prime 7 is excluded because 7 < 16 and 16 = 2 (mod 7).

Cf. A040098, A065903, A065904, A065905, A065906.

a065902(m) = local(n,f,a,j); for(n = 2,m,f = factor(n^4-2); a = matsize(f)[1]; j = 1; while(f[j,1]< = n&&jn,f[j,1],0),","))
a065902(45)

If n^4 - 2 has prime factors > n, then a(n) = smallest of these prime factors, else a(n) = 0.

A065909 First solution mod p of x^4 = 2 for primes p such that more than two solutions exist.

Original entry on oeis.org

18, 5, 27, 28, 35, 46, 131, 48, 252, 104, 45, 123, 51, 9, 69, 77, 51, 177, 472, 261, 55, 117, 224, 562, 12, 264, 273, 132, 127, 500, 17, 197, 107, 36, 206, 671, 127, 159, 137, 684, 329, 564, 316, 314, 197, 98, 661, 925, 461, 170, 930, 151, 1081, 333, 434, 924

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). There are integers which do occur thrice, e.g. 6624. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 27, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 27 is the least one.

Cf. A040098, A007522, A014754, A065910, A065911, A065912.

a065909(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>2,print1(s[1],",")))
a065909(4000)

a(n) = first (least) solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

Definition corrected by Harvey P. Dale, Apr 16 2015

A065910 Second solution mod p of x^4 = 2 for primes p such that more than two solution exists.

Original entry on oeis.org

25, 8, 47, 71, 46, 91, 158, 102, 278, 294, 216, 201, 355, 110, 297, 283, 161, 567, 490, 422, 578, 250, 309, 625, 344, 578, 287, 151, 164, 641, 736, 238, 474, 763, 408, 758, 406, 650, 813, 1090, 1043, 771, 328, 699, 902, 165, 857, 1000, 553, 1148, 1434, 955

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). In this section, there are no integers which do occur thrice. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 47, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 47 is the second one.

Cf. A040098, A007522, A014754, A065909, A065911, A065912.

a065910(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>2,print1(s[2],",")))
a065910(3500)

a(n) = second solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

A065911 Third solution mod p of x^4 = 2 for primes p such that more than two solution exists.

Original entry on oeis.org

48, 81, 66, 162, 211, 190, 179, 251, 299, 299, 385, 416, 526, 827, 736, 766, 936, 586, 703, 779, 639, 999, 980, 808, 1137, 975, 1314, 1458, 1557, 1112, 1041, 1563, 1415, 1150, 1681, 1355, 1723, 1623, 1468, 1303, 1398, 1702, 2265, 1958, 1787, 2668, 2000

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). In this section, there are no integers which do occur thrice. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 66, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 66 is the third one.

Cf. A040098, A007522, A014754, A065909, A065910, A065912.

a065911(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>2,print1(s[3],",")))
a065911(3000)

a(n) = third solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

A065912 Fourth solution mod p of x^4 = 2 for primes p such that more than two solution exists.

Original entry on oeis.org

55, 84, 86, 205, 222, 235, 206, 305, 325, 489, 556, 494, 830, 928, 964, 972, 1046, 976, 721, 940, 1162, 1132, 1065, 871, 1469, 1289, 1328, 1477, 1594, 1253, 1760, 1604, 1782, 1877, 1883, 1442, 2002, 2114, 2144, 1709, 2112, 1909, 2277, 2343, 2492, 2735

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). In this section, there are no integers which do occur thrice. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 86, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 86 is the fourth one.

Cf. A040098, A007522, A014754, A065909, A065910, A065911.

a065912(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>3,print1(s[4],",")))
a065912(3000)

a(n) = fourth solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

A049542 Primes p such that x^10 = 2 has a solution mod p.

Original entry on oeis.org

2, 7, 17, 23, 47, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 193, 199, 223, 233, 239, 241, 257, 263, 313, 337, 353, 359, 367, 383, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 569, 577, 593, 599, 607, 617, 641, 647, 673, 719, 727, 743, 769, 809, 823

Offset: 1

N. J. A. Sloane

Complement of A059263 relative to A000040. - Vincenzo Librandi, Sep 13 2012

			0^10 == 2 (mod 2). 2^10 == 2 (mod 7). 7^10 == 2 (mod 17). 11^10 == 2 (mod 23). 13^10 == 2 (mod 47). 2^10 == 2 (mod 73). 16^10 == 2 (mod 79). 44^10 == 2 (mod 89). 29^10 == 2 (mod 97). - _R. J. Mathar_, Jul 20 2025

R. J. Mathar, Table of n, a(n) for n = 1..1000
Index entries for related sequences

Cf. A000040, A059263.

[p: p in PrimesUpTo(1100) | exists(t){x : x in ResidueClassRing(p) | x^10 eq 2}]; // Vincenzo Librandi, Sep 13 2012

ok[p_]:= Reduce[Mod[x^10- 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

PARI
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/* see A040098 */
```

A040100 Primes p such that x^4 = 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, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 269, 277, 283, 293, 307, 313, 317, 331, 347, 349, 373

Offset: 1

N. J. A. Sloane

Complement of A040098 relative to A000040. - Vincenzo Librandi, Sep 17 2012

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

[p: p in PrimesUpTo(500) | not exists{x : x in ResidueClassRing(p) | x^4 eq 2} ]; // Vincenzo Librandi, Sep 17 2012

ok[p_]:= Reduce[Mod[x^4 - 2, p] == 0, x, Integers] == False;Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 17 2012  *)

A065907 First solution mod p of x^4 = 2 for primes p such that only two solutions exist.

Original entry on oeis.org

2, 8, 15, 17, 15, 3, 48, 4, 16, 34, 33, 47, 98, 92, 68, 63, 114, 78, 153, 157, 107, 36, 156, 115, 86, 58, 222, 297, 57, 6, 18, 235, 66, 142, 221, 395, 227, 33, 120, 408, 368, 131, 301, 408, 253, 149, 318, 405, 459, 121, 30, 206, 122, 28, 543, 472, 88, 283, 696, 246

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three times in this sequence. Confirmed for the first 2399 terms of A007522 (primes < 100000). There are integers which do occur thrice, e.g. 221, 1159.

			a(8) = 4, since 127 is the eighth term of A007522 and 4 is the first solution mod 127 of x^4 = 2.

Cf. A040098, A007522, A065908.

a065907(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2] == 2,print1(s[1],",")))
a065907(1600)

a(n) = first (least) solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has only two solutions mod p, i.e. p is the n-th term of A007522.

cp's OEIS Frontend

A045315 Primes p such that x^8 = 2 has a solution mod p.

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A072936 Primes p that do not divide 2^x+1 for any x>=1.

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A065902 Smallest prime p such that n is a solution mod p of x^4 = 2, or 0 if no such prime exists.

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A065909 First solution mod p of x^4 = 2 for primes p such that more than two solutions exist.

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A065910 Second solution mod p of x^4 = 2 for primes p such that more than two solution exists.

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A065911 Third solution mod p of x^4 = 2 for primes p such that more than two solution exists.

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A065912 Fourth solution mod p of x^4 = 2 for primes p such that more than two solution exists.

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A049542 Primes p such that x^10 = 2 has a solution mod p.

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