| Displaying 1-10 of 10 results found.
page
1
Primes with primitive root 3.
+10
28
2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 139, 149, 163, 173, 197, 199, 211, 223, 233, 257, 269, 281, 283, 293, 317, 331, 353, 379, 389, 401, 449, 461, 463, 487, 509, 521, 557, 569, 571, 593, 607, 617, 631, 641, 653, 677, 691, 701, 739, 751, 773, 797
COMMENTS
To allow primes less than the specified primitive root m (here, 3) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334- A019421. - N. J. A. Sloane, Dec 02 2019 All terms except the first are congruent to 5 or 7 modulo 12. If we define
Pi(N,b) = # {p prime, p <= N, p == b (mod 12)};
Q(N) = # {p prime, 2 < p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,5) + Pi(N,7)), where C = A005596 is Artin's constant. If we further define
Q(N,b) = # {p prime, p <= N, p == b (mod 12), p in this sequence},
then we have:
Q(N,5) ~ (3/5)*Q(N) ~ (12/5)*C*Pi(N,5);
Q(N,7) ~ (2/5)*Q(N) ~ ( 8/5)*C*Pi(N,7).
For example, for the first 1000 terms except for a(1) = 2, there are 593 terms == 5 (mod 12) and 406 terms == 7 (mod 12). (End)
LINKS
J. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, International Conference on Mathematical Computer Engineering (ICMCE-13), pp. 2-7, At VIT University, Chennai, Volume: I, 2013.
MATHEMATICA
pr=3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
PROG
(PARI) isok(p) = isprime(p) && (p!=3) && (znorder(Mod(3, p))+1 == p); \\ Michel Marcus, May 12 2019
Numbers with primitive root 5.
+10
19
2, 3, 6, 7, 9, 14, 17, 18, 23, 27, 34, 37, 43, 46, 47, 49, 53, 54, 73, 74, 81, 83, 86, 94, 97, 98, 103, 106, 107, 113, 137, 146, 157, 162, 166, 167, 173, 193, 194, 197, 206, 214, 223, 226, 227, 233, 243, 257, 263, 274, 277, 283, 289, 293, 307, 314, 317, 334, 343, 346
MATHEMATICA
pr=5; Select[Range[2, 2000], MultiplicativeOrder[pr, # ] == EulerPhi[ # ] &]
CROSSREFS
Cf. A019335 (primes with primitive root 5)
Order of 5 mod n-th prime: least k such that prime(n) divides 5^k-1.
+10
16
1, 2, 0, 6, 5, 4, 16, 9, 22, 14, 3, 36, 20, 42, 46, 52, 29, 30, 22, 5, 72, 39, 82, 44, 96, 25, 102, 106, 27, 112, 42, 65, 136, 69, 37, 75, 156, 54, 166, 172, 89, 15, 19, 192, 196, 33, 35, 222, 226, 114, 232, 119, 40, 25, 256, 262, 67, 27, 276, 140, 282, 292
MATHEMATICA
nn = 5; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]
PROG
(GAP) A000040:=Filtered([1..350], IsPrime);; (PARI) a(n, {base=5}) = my(p=prime(n)); if(base%p, znorder(Mod(base, p)), 0) \\ Jianing Song, May 13 2024
CROSSREFS
Cf. A019335 (full reptend primes in base 5).
Orderly numbers: a number n is orderly if there exists some number k > tau(n) such that the set of the divisors of n is congruent to the set {1,2,...,tau(n)} mod k.
+10
7
1, 2, 5, 7, 8, 9, 11, 12, 13, 17, 19, 20, 23, 27, 29, 31, 37, 38, 41, 43, 47, 52, 53, 57, 58, 59, 61, 67, 68, 71, 72, 73, 76, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 118, 124, 127, 131, 133, 137, 139, 149, 151, 157, 158, 162, 163, 164, 167, 173, 177, 178, 179
COMMENTS
n: {divisors(n)} == {1,2,...,tau(n)} mod k
-------------------------------------------
1: {1} == {1} mod 2
2: {1,2} == {1,2} mod 3
5: {1,5} == {1,2} mod 3
7: {1,7} == {1,2} mod 5
8: {1,2,8,4} == {1,2,3,4} mod 5
9: {1,9,3} == {1,2,3} mod 7
11: {1,11} == {1,2} mod 3 or 9
12: {1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7
13: {1,13} == {1,2} mod 11
17: {1,17} == {1,2} mod 3,5, or 15
19: {1,19} == 1,2 mod 17
20: {1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7
23: {1,23} == {1,2} mod 3,7, or 21
27: {1,27,3,9} == {1,2,3,4} mod 5
29: {1,29} == {1,2} mod 3,9, or 27
31: {1,31} == {1,2} mod 29
37: {1,37} == 1,2 mod 5,7, or 35
38: {1,2,38,19} == {1,2,3,4} mod 5
41: {1,41} == {1,2} mod 3,13, or 39
43: {1,43} == {1,2} mod 41
47: {1,47} == {1,2} mod 3,5,9,15, or 45
52: {1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
53: {1,53} == {1,2} mod 3,17, or 51
57: {1,57,3,19} == {1,2,3,4} mod 5
58: {1,2,58,29} == {1,2,3,4} mod 5
59: {1,59} == {1,2} mod 3,19, or 57
61: {1,61} == {1,2} mod 59
67: {1,67} == {1,2} mod 5,13, or 65
68: {1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
71: {1,71} == {1,2} mod 3,23, or 69
72: {1,2,3,4,18,6,72,8,9,36,24,12} == {1,2,3,4,5,6,7,8,9,10,11,12} mod 13
73: {1,73} == {1,2} mod 71
76: {1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7
79: {1,79} == {1,2} mod 7,11, or 77
83: {1,83} == {1,2} mod 3,9,27, or 81
87: {1,87,3,29} == {1,2,3,4} mod 5
89: {1,89} == {1,2} mod 3,29, or 87
97: {1,97} == {1,2} mod 5,19, or 95
The primes other than 3 are orderly.
Numbers of the form 4p are orderly when p is an odd prime congruent to 3,5, or 6 mod 7.
For primes, k values can be p-2 or a divisor of p-2 other than 1.
T. D. Noe observed that for composite orderly numbers, n, k seems to be one of the three values: tau(n)+1, tau(n)+3, tau(n)+4.
The composite numbers with k = tau(n)+4 are of the form p^2, where prime p == 3 mod 7.
The orderly numbers with k = tau(n)+3 come in many forms. See A168003. It appears that tau(n)+3 is a prime with primitive root 2 ( A001122). The forms for composite orderly numbers with k = tau(n)+1 are too numerous to list here, but seem to occur for any prime k > 3.
Let p be any prime. Then p^(m-2) is in this sequence if m is a prime with primitive root p. For example, 2^(m-2) is here for every m in A001122; 3^(m-2) is here for every m in A019334; 5^(m-2) is here for every m in A019335. For every prime p, there appear to be an infinite number of prime powers p^(m-2) here. All these numbers are actually very orderly ( A167409) because we can choose k = tau(n)+1. - T. D. Noe, Nov 04 2009
EXAMPLE
12 is an orderly number because 12's divisors are 1,2,3,4,6,12 and
1 == 1 (mod 7)
2 == 2 (mod 7)
3 == 3 (mod 7)
4 == 4 (mod 7)
12 == 5 (mod 7)
6 == 6 (mod 7)
MATHEMATICA
orderlyQ[n_] := (For[dd = Divisors[n]; tau = Length[dd]; k = 3, k <= Max[tau + 4, Last[dd] - 2], k++, If[ Union[ Mod[dd, k]] == Range[tau], Return[True]]]; False); Select[ Range[180], orderlyQ] (* Jean-François Alcover, Aug 19 2013 *)
CROSSREFS
Cf. A167409 = very orderly numbers (k = tau(n) + 1). Cf. A167410 = disorderly numbers = numbers not in this sequence. Cf. A167411 = minimal k values for the orderly numbers.
EXTENSIONS
Information about the tau(n)+3 orderly numbers corrected by T. D. Noe, Nov 16 2009
Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(5).
+10
1
1, 2, 6, 16, 22, 36, 42, 46, 52, 72, 82, 96, 102, 106, 112, 136, 156, 166, 172, 192, 196, 222, 226, 232, 256, 262, 276, 282, 292, 306, 316, 346, 352, 372, 382, 396, 432, 442, 462, 466, 502, 522, 546, 556, 562, 576, 586, 592, 606, 612, 616, 646, 652, 672, 676
COMMENTS
Numbers k = p - 1 such that p is a prime with primitive root 5. - Joerg Arndt, Jun 25 2020
MAPLE
filter:= proc(n)
isprime(n+1) and numtheory:-order(5, n+1)=n
end proc:
MATHEMATICA
Select[Prime[Range[1000]], MultiplicativeOrder[5, #] == # - 1&] - 1 (* Jean-François Alcover, Aug 16 2020 *)
PROG
(PARI) forprime(p=2, 10^3, if(p==5, next()); if(znorder(Mod(5, p))==p-1, print1(p-1, ", "))); \\ Joerg Arndt, Jun 25 2020
Multiplicative suborder of 5 (mod n) = sord(5, n).
+10
1
0, 0, 1, 1, 1, 0, 1, 3, 2, 3, 0, 5, 2, 2, 3, 0, 4, 8, 3, 9, 0, 3, 5, 11, 2, 0, 2, 9, 6, 7, 0, 3, 8, 10, 8, 0, 6, 18, 9, 4, 0, 10, 3, 21, 5, 0, 11, 23, 4, 21, 0, 16, 4, 26, 9, 0, 6, 18, 7, 29, 0, 15, 3, 3, 16, 0, 10, 11, 16, 11, 0, 5, 6, 36, 18, 0, 9, 30, 4, 39, 0, 27, 10, 41, 6, 0, 21, 7, 10, 22, 0
COMMENTS
a(n) is minimum e for which 5^e = +/-1 mod n, or zero if no e exists.
For n > 2, a(n) <= (n-1)/2, with equality if (but not only if) n is in A019335. - Robert Israel, Mar 20 2020
REFERENCES
H. Cohen, Course in Computational Algebraic Number Theory, Springer, 1993, p. 25, Algorithm 1.4.3
MAPLE
f:= proc(n) local x;
if n mod 5 = 0 then return 0 fi;
x:= numtheory:-mlog(-1, 5, n);
if x <> FAIL then x else numtheory:-order(5, n) fi
end proc:
f(1):= 0:
MATHEMATICA
Suborder[k_, n_] := If[n > 1 && GCD[k, n] == 1, Min[MultiplicativeOrder[k, n, {-1, 1}]], 0];
a[n_] := Suborder[5, n];
Primes having primitive roots 2, 3, and 5.
+10
1
53, 173, 197, 293, 317, 557, 653, 677, 773, 797, 907, 1277, 1373, 1483, 1493, 1637, 1733, 1747, 1987, 1997, 2083, 2213, 2237, 2333, 2357, 2467, 2477, 2683, 2693, 2837, 2957, 3307, 3413, 3533, 3547, 3557, 3643, 3677, 3797, 3917, 4003, 4013, 4133, 4157
MATHEMATICA
fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[600]], fQ[2, #] && fQ[3, #] && fQ[5, #] &]
Select[Prime[Range[600]], SequenceCount[PrimitiveRootList[#], {2, 3, 5}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 03 2018 *)
Primes having primitive roots 2, 3, 5, and 7.
+10
1
173, 293, 677, 773, 797, 907, 1277, 1637, 1747, 2083, 2357, 2477, 2693, 2957, 3533, 3797, 4133, 4157, 4373, 4493, 4603, 4637, 4877, 4973, 5333, 5477, 5717, 5813, 5923, 6053, 6173, 6317, 6547, 6653, 6763, 7013, 7517, 8237, 8573, 8693, 8837, 9173, 9533
MATHEMATICA
fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[1200]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] &]
Select[Prime[Range[1200]], SubsetQ[PrimitiveRootList[#], {2, 3, 5, 7}]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 16 2020 *)
Primes having primitive roots 2, 3, 5, 7, 11, and 13.
+10
1
293, 2477, 4373, 6173, 7013, 9173, 9677, 10853, 13037, 13397, 13613, 13877, 14957, 15413, 17093, 17597, 18413, 18917, 19157, 22277, 22613, 24317, 26813, 27653, 27893, 29333, 30197, 31517, 33893, 34613, 34877, 35573, 37253, 40493, 41117, 41333, 42437
MATHEMATICA
fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[4500]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] &]
Primes having primitive roots 2, 3, 5, 7, 11, 13, and 17.
+10
1
2477, 9173, 10853, 13877, 14957, 15413, 22277, 22613, 24317, 27653, 30197, 34877, 37253, 41117, 41333, 42437, 42677, 43973, 48677, 51413, 55733, 61613, 62597, 63773, 66293, 72533, 73757, 74093, 76733, 79397, 79757, 82997, 86357, 90173, 92237, 92333, 95597
MATHEMATICA
fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[10000]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] && fQ[17, #] &]
Search completed in 0.009 seconds
| 
