A040028 -id:A040028 - cp's OEIS frontend

A059940 Smallest prime p such that x = n is a solution mod p of x^3 = 2, or 0 if no such prime exists.

3, 5, 31, 41, 107, 11, 17, 727, 499, 443, 863, 439, 457, 3373, 23, 1637, 53, 6857, 31, 47, 5323, 811, 6911, 919, 29, 19681, 439, 739, 13499, 29789, 43, 7187, 43, 461, 23327, 50651, 59, 2579, 2909, 22973, 2179, 15901, 14197, 293, 1187, 34607, 11059

Offset: 2

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^3 = 2 iff n^3-2 has a prime factor > n; n is a solution mod p of x^3 = 2 iff p is a prime factor of n^3-2 and p > n.

n^3-2 has at most two prime factors > n, consequently these factors are the only primes p such that n is a solution mod p of x^3 = 2. For n such that n^3-2 has no prime factor > n (the zeros in the sequence; they occur beyond the last entry shown in the database) see A060591. For n such that n^3-2 has two prime factors > n, cf. A060914.

			a(2) = 3, since 2 is a solution mod 3 of x^3 = 2 and 2 is not a solution mod p of x^3 = 2 for prime p = 2. Although 2^3 = 2 mod 2, prime 2 is excluded because 0 < 2 and 2 = 0 mod 2. a(5) = 41, since 5 is a solution mod 41 of x^3 = 2 and 5 is not a solution mod p of x^3 = 2 for primes p < 41. Although 5^3 = 2 mod 3, prime 3 is excluded because 3 < 5 and 5 = 2 mod 3.

Cf. A040028, A060121, A060122, A060123, A060124, A060591, A060914.

If n^3-2 has prime factors > n, then a(n) = least of these prime factors, else a(n) = 0.

A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists.

Original entry on oeis.org

0, 2, 3, 7, 8, 16, 26, 5, 21, 18, 38, 49, 50, 16, 26, 6, 81, 54, 98, 70, 157, 161, 58, 147, 21, 86, 92, 197, 50, 249, 137, 184, 119, 45, 45, 261, 198, 61, 176, 143, 51, 103, 221, 72, 11, 219, 35, 86, 385, 384, 141, 143, 225, 92, 245, 533, 557, 473, 170, 375, 516

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. For i > 1, i is a solution mod p of x^3 = 2 iff p is a prime factor of i^3-2 and p > i (cf. comment to A059940). i^3-2 has at most two prime factors > i. Hence i is a solution mod p of x^3 = 2 for at most two different p and therefore no integer occurs more than twice in this sequence. There are integers which do occur twice, e.g. 16, 21, 26 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(9) = 21, since 47 is the ninth term of A045309 and 21 is the only solution mod 47 of x^3 = 2.

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

Cf. A040028, A045309, A059940, A060914, A060122, A060123, A060124.

Res:=0,2: count:= 2: p:= 3:
while count < 100 do
p:= nextprime(p);
   if p mod 3 = 2 then
    count:= count+1;
    Res:= Res, numtheory:-mroot(2,3,p);
fi
od:
Res; # Robert Israel, Sep 12 2018

terms = 100;
A045309 = Select[Prime[Range[2 terms]], Mod[#, 3] != 1&];
a[n_] := PowerMod[2, 1/3, A045309[[n]]];
Array[a, terms] (* Jean-François Alcover, Feb 27 2019 *)

a(n) = first (only) solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has only one solution mod p, i.e. p is the n-th term of A045309.

A060122 Smallest solution mod p of x^3 = 2 for primes p such that more than one solution exists.

Original entry on oeis.org

4, 20, 57, 32, 62, 68, 52, 152, 120, 52, 53, 72, 13, 14, 10, 54, 61, 94, 9, 339, 29, 23, 25, 114, 159, 131, 469, 206, 178, 892, 628, 162, 544, 709, 647, 799, 49, 57, 709, 218, 1118, 585, 858, 332, 528, 119, 1151, 1024, 152, 798, 42, 235, 71, 535, 733, 257, 228

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. No integer occurs more than twice in this sequence (cf. comment to A060121). There are integers which do occur twice, e.g. 52, 57, 152 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(3) = 57, since 109 is the third term of A014752, 57, 58 and 103 are the solutions mod 109 of x^3 = 2 and 57 is the least one.

Cf. A040028, A014752, A059940, A060914, A060121, A060123, A060124.

a(n) = first (least) solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has more than one solution mod p, i.e. p is the n-th term of A014752.

A060123 Second solution mod p of x^3 = 2 for primes p such that more than one solution exists.

Original entry on oeis.org

7, 32, 58, 100, 116, 179, 79, 181, 186, 270, 130, 394, 28, 34, 97, 94, 73, 288, 348, 407, 298, 231, 381, 125, 315, 458, 781, 385, 425, 928, 1095, 362, 1186, 992, 1046, 1053, 116, 542, 1236, 425, 1129, 1259, 1344, 1553, 570, 200, 1328, 1286, 888, 1433, 808

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. No integer occurs more than twice in this sequence (cf. comment to A060121). There are integers which do occur twice, e.g. 116, 425 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(3) = 58, since 109 is the third term of A014752 and 58 is the second solution mod 109 of x^3 = 2.

Cf. A040028, A014752, A059940, A060914, A060121, A060122, A060124.

a(n) = second solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has more than one solution mod p, i.e. p is the n-th term of A014752.

A060124 Third solution mod p of x^3 = 2 for primes p such that more than one solution exists.

Original entry on oeis.org

20, 34, 103, 122, 136, 199, 98, 221, 260, 292, 214, 400, 398, 409, 392, 453, 509, 309, 370, 720, 412, 557, 513, 758, 547, 462, 888, 502, 724, 978, 1123, 935, 1212, 1457, 1501, 1402, 1492, 1100, 1501, 1110, 1307, 1734, 1400, 1777, 835, 1680, 1555, 1868

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. No integer occurs more than twice in this sequence (cf. comment to A060121). There are integers which do occur twice, e.g. 1501 (cf. A060914). Moreover, no integer occurs more than twice in A060121, A060122, A060123 and A060124 taken together.

			a(3) = 103, since 109 is the third term of A014752 and 103 is the third solution mod 109 of x^3 = 2.

Cf. A040028, A014752, A059940, A060914, A060121, A060122, A060123.

a(n) = third solution mod p of x^3 = 2, where p is the n-th prime such that x^3 = 2 has more than one solution mod p, i.e. p is the n-th term of A014752.

A065903 Integers i > 1 for which there is no prime p such that i is a solution mod p of x^4 = 2.

Original entry on oeis.org

1689, 1741, 3306, 3894, 4362, 4587, 4999, 5754, 6025, 6371, 6668, 7012, 7982, 9054, 9158, 9695, 9742, 9832, 10056, 10664, 11005, 12027, 12385, 13676, 13895, 14026, 14059, 16104, 16239, 16903, 17050, 17153, 18079, 18202, 18642, 20349, 21060

Offset: 1

Klaus Brockhaus, Nov 28 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has one resp. two resp. three prime factors > i; cf. A065904 resp. A065905 resp. A065906.

			a(2) = 1741, since 1741 is (after 1689) the second integer i for which there are no primes p > i such that i is a solution mod p of x^4 = 2, or equivalently, 1741^4 - 2 = 9187452028559 = 7*7*79*887*1609*1663 has no prime factor > 1741. (cf. A065902).

Cf. A040028, A065902, A065904, A065905, A065906.

a065903(m) = local(c,n,f,a); c = 0; n = 2; while(c

a(n) = n-th integer i such that i^4 - 2 has no prime factor > i.

A040034 Primes p such that x^3 = 2 has no solution mod p.

Original entry on oeis.org

7, 13, 19, 37, 61, 67, 73, 79, 97, 103, 139, 151, 163, 181, 193, 199, 211, 241, 271, 313, 331, 337, 349, 367, 373, 379, 409, 421, 463, 487, 523, 541, 547, 571, 577, 607, 613, 619, 631, 661, 673, 709, 751, 757, 769, 787, 823, 829, 853, 859, 877, 883, 907, 937

Offset: 1

N. J. A. Sloane

Primes represented by the quadratic form 4x^2 + 2xy + 7y^2, whose discriminant is -108. - T. D. Noe, May 17 2005

Complement of A040028 relative to A000040. - Vincenzo Librandi, Sep 17 2012

			A cube modulo 7 can only be 0, 1 or 6, but not 2, hence the prime 7 is in the sequence.
Because x^3 = 2 mod 11 when x = 7 mod 11, the prime 11 is not in the sequence.

Klaus Brockhaus, Table of n, a(n) for n=1..1000
David Bernier, A strong primality test based on third-order linear recurrences, ResearchGate (2025). See p. 8
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Bishnu Paudel and Chris Pinner, The integer group determinants for the abelian groups of order 18, arXiv:2412.10638 [math.NT], 2024. See p. 3.

[ p: p in PrimesUpTo(937) | forall(t){x : x in ResidueClassRing(p) | x^3 ne 2} ]; // Klaus Brockhaus, Dec 05 2008

insolublePrimeQ[p_]:= Reduce[Mod[x^3 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[200]], insolublePrimeQ] (* Vincenzo Librandi Sep 17 2012 *)

forprime(p=2,10^3,if(#polrootsmod(x^3-2,p)==0,print1(p,", "))) \\ Joerg Arndt, Jul 16 2015

More terms from Klaus Brockhaus, Dec 05 2008

A060914 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^3 = 2.

Original entry on oeis.org

7, 16, 20, 21, 26, 32, 34, 45, 49, 50, 52, 54, 57, 58, 61, 70, 72, 79, 81, 86, 92, 94, 98, 103, 111, 112, 114, 116, 119, 122, 125, 130, 136, 137, 141, 143, 147, 152, 157, 160, 170, 176, 179, 181, 184, 186, 197, 198, 199, 214, 221, 222, 225, 231, 234, 236, 240

Offset: 1

Klaus Brockhaus, Apr 08 2001

nonn

Solutions mod p are represented by integers from 0 to p - 1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^3 = 2 iff i^3 - 2 has a prime factor > i; i is a solution mod p of x^3 = 2 iff p is a prime factor of i^3 - 2 and p > i. i^3 - 2 has at most two prime factors > i. For i such that i^3 - 2 has no prime factors > i; cf. A060591.

			a(3) = 20, since 20 is (after 7 and 16) the third integer i for which there are two primes p > i (viz. 31 and 43) such that i is a solution mod p of x^3 = 2, or equivalently, 20^3 - 2 = 7998 = 2*3*31*43 has two prime factors > 20. (cf. A059940).

A040028, A059940, A060591.

a(n) = n-th integer i such that i^3 - 2 has two prime factors > i.

A065904 Integers i > 1 for which there is one prime p such that i is a solution mod p of x^4 = 2.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, 26, 29, 31, 32, 37, 38, 39, 40, 41, 42, 43, 44, 49, 50, 52, 53, 54, 59, 60, 61, 62, 64, 65, 70, 72, 73, 74, 75, 77, 79, 80, 82, 83, 85, 87, 89, 93, 94, 95, 96, 97, 99, 100, 101, 103, 108, 109, 111, 116, 119, 121

Offset: 1

Klaus Brockhaus, Nov 28 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. two resp. three prime factors > i; cf. A065903 resp. A065905 resp. A065906.

			a(3) = 4, since 4 is (after 2 and 3) the third integer i for which there is one prime p > i (viz. 127) such that i is a solution mod p of x^4 = 2, or equivalently, 4^4 - 2 = 254 = 2*127 has one prime factor > 4 (cf. A065902).

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

Cf. A040028, A065902, A065903, A065905, A065906.

filter:= n -> nops(select(`>`, numtheory:-factorset(n^4-2),n))=1:
select(filter, [$2..1000]); # Robert Israel, Jan 30 2017

okQ[n_] := Length[Select[FactorInteger[n^4 - 2][[All, 1]], # > n&]] == 1;
Select[Range[2, 200], okQ] (* Jean-François Alcover, Mar 26 2019, after Robert Israel *)

a065904(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(cn,s = concat(s,f[j,1]))); if(matsize(s)[2] == 1,print1(n,","); c++); n++)
a065904(70)

a(n) = n-th integer i such that i^4 - 2 has one prime factor > i.

A065905 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^4 = 2.

Original entry on oeis.org

5, 8, 16, 17, 18, 25, 27, 28, 30, 33, 34, 35, 36, 45, 46, 47, 51, 56, 57, 58, 63, 66, 67, 68, 69, 71, 76, 78, 81, 84, 86, 88, 90, 91, 92, 98, 102, 104, 105, 106, 107, 110, 112, 113, 114, 115, 117, 118, 120, 122, 123, 125, 126, 127, 131, 132, 133, 134, 135, 136, 137

Offset: 1

Klaus Brockhaus, Nov 28 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. one resp. three prime factors > i cf. A065903 resp. A065904 resp. A065906.

			a(3) = 16, since 16 is (after 5 and 8) the third integer i for which there are two primes p > i (viz. 31 and 151) such that i is a solution mod p of x^4 = 2, or equivalently, 16^4 - 2 = 65534 = 2*7*31*151 has two prime factors > 4. (cf. A065902).

Cf. A040028, A065902, A065903, A065904, A065906.

a065905(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(cn,s = concat(s,f[j,1]))); if(matsize(s)[2] == 2,print1(n,","); c++); n++)
a065905(65)

a(n) = n-th integer i such that i^4 - 2 has two prime factors > i.

cp's OEIS Frontend

A059940 Smallest prime p such that x = n is a solution mod p of x^3 = 2, or 0 if no such prime exists.

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A060121 First solution mod p of x^3 = 2 for primes p such that only one solution exists.

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A060122 Smallest solution mod p of x^3 = 2 for primes p such that more than one solution exists.

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A060123 Second solution mod p of x^3 = 2 for primes p such that more than one solution exists.

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A060124 Third solution mod p of x^3 = 2 for primes p such that more than one solution exists.

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A065903 Integers i > 1 for which there is no prime p such that i is a solution mod p of x^4 = 2.

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PARI

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A040034 Primes p such that x^3 = 2 has no solution mod p.

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Magma

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A060914 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^3 = 2.

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A065904 Integers i > 1 for which there is one prime p such that i is a solution mod p of x^4 = 2.

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