|
|
A309588
|
|
Primes p with 4 zeros in a fundamental period of A006190 mod p.
|
|
12
|
|
|
5, 13, 29, 37, 41, 73, 89, 97, 109, 137, 149, 157, 181, 193, 197, 229, 233, 241, 269, 281, 293, 317, 349, 353, 373, 389, 397, 401, 409, 421, 449, 457, 461, 509, 541, 557, 577, 593, 613, 617, 653, 661, 701, 709, 733, 761, 769, 773, 797, 821, 853, 857, 877
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
For p > 2, p is in this sequence if and only if (all these conditions are equivalent):
(b) ord(p,(3+sqrt(13))/2) == 4 (mod 8), where ord(p,u) is the smallest integer k > 0 such that (u^k - 1)/p is an algebraic integer;
(c) ord(p,(11+3*sqrt(13))/2) == 2 (mod 4);
(e) ord(p,-(11+3*sqrt(13))/2) is odd.
In general, let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let pi(k) be the Pisano period of {x(n)} modulo k, i.e., pi(k) = min{l > 0 : x(n+l) == x(n) (mod k) for all n}, r(k) = min{l > 0 : k divides x(l)} and w(k) be the number of zeros in a fundamental period of {x(n)} modulo k. Let u = (m + sqrt(m^2+4))/2, p be an odd prime, then these conditions are equivalent:
(1) w(p) = 4;
(2) pi(p) == 4 (mod 8);
(3) ord(p,u) == 4 (mod 8);
(4) ord(p,u^2) == 2 (mod 4);
(5) r(p) is odd;
(6) ord(p,-u^2) is odd.
This can be shown by noting that pi(p) = p^c*ord(p,u) and r(p) = p^c*ord(p,-u^2) for some c (if p does not divide m^2 + 4 then c = 0, otherwise c = 1). Also, Pi(p) is always even, so ord(p,u) is always even.
This sequence contains all primes congruent to 5, 21, 33, 37, 41, 45 modulo 52.
Conjecturely, this sequence has density 1/3 in the primes.
|
|
LINKS
|
|
|
PROG
|
(PARI) forprime(p=2, 900, if(A322906(p)==4, print1(p, ", ")))
|
|
CROSSREFS
|
Let {x(n)} be the sequence defined in the comment section.
| m=1 | m=2 | m=3
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|