A014752 -id:A014752 - cp's OEIS frontend

A059899 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is p.

31, 229, 397, 439, 457, 499, 601, 643, 691, 727, 739, 811, 919, 997, 1021, 1051, 1093, 1327, 1459, 1657, 1699, 1753, 1933, 1999, 2113, 2179, 2203, 2251, 2281, 2341, 2347, 2383, 2671, 2731, 2767, 2791, 2833, 2953, 2971, 3061, 3229, 3259, 3331, 3373, 3391

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Subsequence of A040028 and of A014752, complement of A059914 relative to A014752. Solutions mod p are represented by integers from 0 to p-1.

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

Cf. A040028, A014752, A059914.

filter:= proc(p) local S;
  if not isprime(p) then return false fi;
  S:= map(t -> rhs(t[1]), [msolve(x^3=2,p)]);
  nops(S) = 3 and convert(S,`+`) = p
end proc:
select(filter, [seq(i,i=7..5000, 6)]); # Robert Israel, Aug 13 2024

A059914 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is 2*p.

Original entry on oeis.org

43, 109, 127, 157, 223, 277, 283, 307, 433, 733, 1069, 1399, 1423, 1471, 1579, 1597, 1627, 1723, 1777, 1789, 1801, 1831, 2017, 2089, 2143, 2287, 2689, 2749, 2917, 3163, 3181, 3271, 3343, 3541, 3607, 3631, 3823, 3889, 4057, 4129, 4153, 4177, 4339, 4513

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Subsequence of A040028 and of A014752, complement of A059899 relative to A014752. Solutions mod p are represented by integers from 0 to p-1.

Cf. A040028, A014752, A059899, A001133.

A227622 Primes p of the form m^2 + 27.

Original entry on oeis.org

31, 43, 127, 223, 283, 811, 1051, 1471, 1627, 2143, 2731, 3163, 3391, 4651, 5503, 6427, 8863, 9631, 16411, 16927, 18523, 23131, 23743, 27583, 28927, 29611, 33151, 37663, 42463, 43291, 44971, 45823, 56671, 65563, 70783, 78427, 80683, 84127, 87643, 106303, 110251, 122527, 123931, 131071

Offset: 1

William P. Orrick, Jul 17 2013

Orders for which residues mod p of the form x^i, i congruent to 0, 1, or 3 (mod 6), form a difference set with parameters (v,k,lambda)=(p,(p-1)/2,(p-3)/4), where x is a primitive root such that 3=x^j, with j congruent to 1 (mod 6). This construction is due to Marshall Hall. Such a difference set has the same parameters as the difference set formed by quadratic residues, that is, the Paley difference set, but is not equivalent to it. Both difference sets give rise to Hadamard matrices of size p+1.
From Peter Bala, Nov 19 2021: (Start)
2 is a cube mod p (a particular case of a more general result of Gauss). See A014752.
Primes of the form a^2 + 6*a + 36, where a is an integer.
Let p == 1 (mod 6) be a prime. There are integers c and d, unique up to sign, such that 4*p = c^2 + 27*d^2 [see, for example, Ireland and Rosen, Proposition 8.3.2]. This sequence lists those primes with d = 2. Cf. A005471 (case d = 1) and A349461 (case d = 3).
Primes p of the form m^2 + 27 are related to cyclic cubic fields in several ways:
(1) The cubic polynomial X^3 - p*X + 2*p, with discriminant 4*m^2*p^2, a square, is irreducible over Q by Eisenstein's criteria. It follows that the Galois group of the polynomial over Q is the cyclic group C_3 (apply Conrad, Corollary 2.5).
Note that the roots of the cubic X^3 - p*X + 2*p, are the differences n_0 - n_1, n_1 - n_2 and n_2 - n_0 of the cubic Gaussian periods n_i for the modulus p.
(2) The cubic 2*X^3 + p*(X + 2)^2, with discriminant 64*m^2*p^2, a square, is irreducible over Q by Eisenstein's criteria, and so the Galois group of the polynomial over Q is the cyclic group C_3.
(3) The cubic X^3 - (m-3)*X^2 - 2*(m+3)*X - 8, has discriminant (2*p)^2, a square. (This is the polynomial g_3(m-3, 0, -2; X) in the notation of Hashimoto and Hoshi.) The cubic is irreducible over Q for nonzero m by the Rational Root Theorem and hence the Galois group of the polynomial over Q is the cyclic group C_3. (End)

			For p=31, using x=3 as primitive root, the set of residues {1,2,3,4,6,8,12,15,16,17,23,24,27,29,30} is a difference set.
2 a cube mod p: 4^3 == 2 (mod 31); 20^3 == 2 (mod 43); 8^3 == 2 (mod 127); 68^3 == 2 (mod 223). - _Peter Bala_, Nov 19 2021

K. Ireland and M. Rosen, A classical introduction to modern number theory, vol. 84, Graduate Texts in Mathematics. Springer-Verlag. [Prop. 8.3.2, p. 96]
Thomas Storer, Cyclotomy and difference sets. Markham, Chicago, 1967, pages 73-76.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Bala, Notes on the period polynomial for the cubic Gaussian periods
Keith Conrad, Galois groups of cubics and quartics (not in characteristic 2)
Marshall Hall Jr., A survey of difference sets, Proc. Amer. Math. Soc. 7 (1956) 975-986.
Ki-Ichiro Hashimoto and Akinari Hoshi, Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations, Math. Comp., Vol. 74, No. 251, 2005, pp. 1519-1530
D. H. Lehmer and Emma Lehmer, The Lehmer Project, Math. of Comp., Vol. 61, No. 203, 1993, pp. 313-317.

Cf. A005471, A014752, A349461.

Mathematica
```
Select[Table[m^2+27,{m,0,100}],PrimeQ]
```

for(n=0,10^3,my(p=n^2+27);if(isprime(p),print1(p,", "))); \\ Joerg Arndt, Jul 18 2013

A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

Original entry on oeis.org

274177, 319489, 6700417, 825753601, 1214251009, 6487031809, 646730219521, 6597069766657, 25409026523137, 31065037602817, 46179488366593, 151413703311361, 231292694251438081, 1529992420282859521, 2170072644496392193, 3603109844542291969

Offset: 1

Arkadiusz Wesolowski, Feb 07 2022

nonn

Subsequence of A014752.

			a(1) = 503^2 + 27*28^2 = 274177 is a prime factor of 2^(2^6) + 1;
a(2) = 383^2 + 27*80^2 = 319489 is a prime factor of 2^(2^11) + 1;
a(3) = 887^2 + 27*468^2 = 6700417 is a prime factor of 2^(2^5) + 1;
a(4) = 27017^2 + 27*1884^2 = 825753601 is a prime factor of 2^(2^16) + 1;
a(5) = 2561^2 + 27*6688^2 = 1214251009 is a prime factor of 2^(2^15) + 1;

Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.

Cf. A002476, A014752, A023394, A204620, A229850, A229853.

isok(p) = if(p%6==1 && isprime(p), my(z=znorder(Mod(2, p))); z>>valuation(z, 2)==1, return(0));

A002476 INTERSECT A023394.

A016108 Numbers k=3*m+1 such that 2^m == 1 (mod k).

Original entry on oeis.org

1, 31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691, 727, 733, 739, 811, 919, 997, 1021, 1051, 1069, 1093, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1723, 1729, 1753, 1777, 1789, 1801, 1831, 1933, 1999, 2017, 2047, 2089, 2113, 2143, 2179, 2203

Offset: 1

Max Alekseyev, Jun 23 2012

nonn

Prime terms are listed in A014752.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A014752.

Join[{1},Select[Table[3m+1,{m,1000}],PowerMod[2,(#-1)/3,#]==1&]] (* Harvey P. Dale, Apr 26 2016 *)

A360652 Primes of the form x^2 + 432*y^2.

Original entry on oeis.org

433, 457, 601, 1657, 1753, 1777, 1801, 2017, 2089, 2113, 2281, 2689, 2833, 2953, 3457, 3889, 4057, 4129, 4153, 4177, 4513, 4657, 4729, 5113, 5209, 5449, 5569, 5737, 5953, 6217, 6361, 6673, 6961, 7057, 7321, 7369, 7537, 7753, 7873, 8353, 8377, 8713, 8761, 8929

Offset: 1

Arkadiusz Wesolowski, Feb 15 2023

nonn

Supersequence of A351332. Thus every prime congruent to 1 mod 3 that divides a Fermat number is in this sequence.

Every Fermat number that is a semiprime has a prime of this form as a factor.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A014752, A023394, A229850, A351332.

[p: p in PrimesUpTo(8929) | NormEquation(432, p) eq true];

select(p->my(m=Mod(2, p)^(p\12)); p>11 && (m==1||m==p-1), primes(1110))

A306787 Prime numbers p such that there exists an integer k such that p-1 does not divide k-1 and x -> x + x^k is a bijection from Z/pZ to Z/pZ.

Original entry on oeis.org

31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691, 727, 733, 739, 811, 919, 997, 1021, 1051, 1069, 1093, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1723, 1753, 1777, 1789, 1801, 1831, 1933, 1999, 2017

Offset: 1

Elias Caeiro, Apr 16 2019

nonn

If x -> x + x^k is a bijection from Z/pZ to Z/pZ then the following facts hold:
-v_2(k-1) >= v_2(p-1)
-gcd(k+1,p-1) = 2
-2^(k-1) = 1 (mod p).
The third fact is very important as it shows that for a given k there are a finite number of solutions p.
If p = 1 (mod 3) and 2^((p-1)/3) = 1 then either k = (p-1)/3+1 or k = 2*(p-1)/3+1 has the wanted property (see sequence A014752 for more information when this happens). It is a sufficient but not necessary condition since 3251 also appears in this sequence but 3 does not divide 3250.

			For p = 31 and k = 21, x -> x + x^k is a bijection.

Elias Caeiro, Table of n, a(n) for n = 1...212
Problèmes du 9ème Tournoi Français des Jeunes Mathématiciennes et Mathématiciens, Problem 7 question 7, 2019 (in French).

Cf. A014752.

A321867 Numbers k such that 8k+1, 12k+1 and 24k+1 are primes and the last two are also of the form x^2 + 27y^2, so the tetrahedral number T(24k+1) is a Fermat pseudoprime to base 2.

Original entry on oeis.org

1179, 1274, 1895, 4775, 5304, 5874, 6525, 6639, 13035, 16380, 17424, 18459, 21239, 21584, 21714, 22475, 22715, 22734, 27410, 28304, 29340, 29909, 31755, 32294, 34700, 37700, 41525, 42164, 42929, 42950, 43275, 46415, 47174, 47300, 53364, 57879, 59739, 61194

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

The first 3 terms were found by Rotkiewicz.

The generated tetrahedral pseudoprimes are 3776730328549, 4765143438329, 15680770945781, ...

			1179 is in the sequence since 8*1179+1 = 9433, 12*1179+1 = 14149 = 107^2 + 27*10^2 and 24*1179+1 = 28297 = 163^2 + 27*8^2 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

Cf. A000292, A001567, A014752, A321866.

sqQ[n_] := n>0 && IntegerQ[Sqrt[n]]; sqsumQ[n_] := PrimeQ[n] && False =!= Reduce[ x^2 + 27 y^2 == n, {x, y}, Integers]; aQ[n_] := PrimeQ[8n+1] && sqsumQ[12n+1] && sqsumQ[24n+1]; Select[Range[100000], aQ]

A256172 Primes of the form 6p + 1 with p prime that are also of the form x^2 + 27y^2 and congruent to 7 mod 24.

Original entry on oeis.org

31, 223, 439, 1399, 2383, 2767, 3343, 3463, 3607, 4567, 6079, 7927, 8167, 8287, 8719, 10159, 10663, 11959, 14503, 15559, 15727, 17383, 18223, 19087, 20743, 21487, 21559, 24007, 25639, 26647, 27103, 27583, 28807, 28879, 29167, 29599, 31183, 32359, 33343

Offset: 1

Arkadiusz Wesolowski, Jun 01 2015

nonn

a(n) divides 2^m - 1, where m = (a(n) - 7)/6 + 1.

Subsequence of A122094.

Cf. A014752, A051644, A107006.

A014752 INTERSECT A051644 INTERSECT A107006.

A347035 Primes whose field of roots of unity contains a cubic subfield.

Original entry on oeis.org

31, 43, 109, 127, 157, 189

Offset: 1

Michel Marcus, Aug 12 2021

This is probably an erroneous version of A014752; last term 189 is not prime.

Fernando Q. Gouvêa and Jonathan Webster, Kurt Hensel on Common Inessential Discriminant Divisors, 1894, arXiv:2108.05327 [math.HO], 2021. See p. 36.
K. Hensel, Arithmetische Untersuchungen über die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung, Journal für die reine und angewandte Mathematik (1894), Volume: 113, page 128-160. See p. 148.

Cf. A014752.

cp's OEIS Frontend

A059899 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is p.

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A059914 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is 2*p.

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A227622 Primes p of the form m^2 + 27.

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A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

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A016108 Numbers k=3*m+1 such that 2^m == 1 (mod k).

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A360652 Primes of the form x^2 + 432*y^2.

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Magma

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A306787 Prime numbers p such that there exists an integer k such that p-1 does not divide k-1 and x -> x + x^k is a bijection from Z/pZ to Z/pZ.

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A321867 Numbers k such that 8k+1, 12k+1 and 24k+1 are primes and the last two are also of the form x^2 + 27y^2, so the tetrahedral number T(24k+1) is a Fermat pseudoprime to base 2.

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A256172 Primes of the form 6*p + 1 with p prime that are also of the form x^2 + 27*y^2 and congruent to 7 mod 24.

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A256172 Primes of the form 6p + 1 with p prime that are also of the form x^2 + 27y^2 and congruent to 7 mod 24.