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A058853
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Primes p such that x^43 = 2 has a solution mod p.
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6
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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277
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OFFSET
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1,1
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COMMENTS
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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]
Differs from A000040 - the prime 173 does not appear.
For case x^31 = 2 the first missing prime is 311 (64th term).
For case x^47 = 2 the first missing prime is 283 (61st term).
For case x^59 = 2 the first missing prime is 709 (127th term).
For case x^61 = 2 the first missing prime is 367 (73rd term).
It is conjectured that this sequence has density 42/43 ~ 0.976744 over all the primes.
N | # of terms among
| the first N primes
------+--------------------
10^4 | 9758
10^5 | 97681
10^6 | 976798
10^7 | 9767551
10^8 | 97674723
If the conjecture is correct, then a(n) ~ 43/42 * n log n.
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)
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
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LINKS
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MATHEMATICA
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ok[p_]:= Reduce[Mod[x^43 - 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[1000]], ok] (* Vincenzo Librandi Sep 14 2012 *)
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PROG
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(Magma) [p: p in PrimesUpTo(400) | exists(t){x : x in ResidueClassRing(p) | x^43 eq 2}]; // Vincenzo Librandi Sep 14 2012
(PARI) isA058853(p) = isprime(p) && ispower(Mod(2, p), 43) \\ Jianing Song, Mar 07 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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