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A058853 Primes p such that x^43 = 2 has a solution mod p.

%I A058853 #39 Sep 08 2022 08:45:02
%S A058853 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
%T A058853 97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,179,181,
%U A058853 191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277
%N A058853 Primes p such that x^43 = 2 has a solution mod p.
%C A058853 Primes not of the form 43k + 1. - _Charles R Greathouse IV_, Aug 22 2011 [Not so! The smallest counterexample is 5419: 5419 = 43*126 + 1, but 2^43 == 2 (mod 5419), so it is here. - _Jianing Song_, Mar 07 2021]
%C A058853 Differs from A000040 - the prime 173 does not appear.
%C A058853 For case x^31 = 2 the first missing prime is 311 (64th term).
%C A058853 For case x^47 = 2 the first missing prime is 283 (61st term).
%C A058853 For case x^59 = 2 the first missing prime is 709 (127th term).
%C A058853 For case x^61 = 2 the first missing prime is 367 (73rd term).
%C A058853 Complement of A059243 relative to A000040. - _Vincenzo Librandi_, Sep 14 2012
%C A058853 From _Jianing Song_, Mar 07 2021: (Start)
%C A058853 It is conjectured that this sequence has density 42/43 ~ 0.976744 over all the primes.
%C A058853      N  | # of terms among
%C A058853         | the first N primes
%C A058853   ------+--------------------
%C A058853    10^4 |          9758
%C A058853    10^5 |         97681
%C A058853    10^6 |        976798
%C A058853    10^7 |       9767551
%C A058853    10^8 |      97674723
%C A058853 If the conjecture is correct, then a(n) ~ 43/42 * n log n.
%C A058853 In general, let p be a prime, a be an integer that is not a p-th power, then it seems that the density of prime factors of x^p - a over all the primes is 1 - 1/p. This is well-known to be correct for p = 2. (End)
%C A058853 The generalized conjecture above is equivalent to: let P(p,1) be the set of primes congruent to 1 modulo p, P(p,1;a) be the set of primes q congruent to 1 modulo p such that x^p == a (mod q) has a solution, where p is a prime, a is not a p-th power, then the density of P(p,1;a) over P(p,1) is 1/p. - _Jianing Song_, Mar 09 2021
%H A058853 R. J. Mathar, <a href="/A058853/b058853.txt">Table of n, a(n) for n = 1..1000</a>
%H A058853 <a href="/index/Pri#smp">Index entries for related sequences</a>
%t A058853 ok[p_]:= Reduce[Mod[x^43 - 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[1000]], ok]  (* _Vincenzo Librandi_ Sep 14 2012 *)
%o A058853 (Magma) [p: p in PrimesUpTo(400) | exists(t){x : x in ResidueClassRing(p) | x^43 eq 2}]; // _Vincenzo Librandi_ Sep 14 2012
%o A058853 (PARI) isA058853(p) = isprime(p) && ispower(Mod(2,p), 43) \\ _Jianing Song_, Mar 07 2021
%K A058853 nonn,easy
%O A058853 1,1
%A A058853 _Patrick De Geest_, Dec 15 2000
%E A058853 The old formula "a(n) ~ 42/41 * n log n" based on false observation from _Charles R Greathouse IV_, Aug 22 2011 removed by _Jianing Song_, Mar 07 2021