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A040117 Primes congruent to 5 (mod 12). Also primes p such that x^4 = 9 has no solution mod p.

%I A040117 #54 Jul 08 2025 22:48:40
%S A040117 5,17,29,41,53,89,101,113,137,149,173,197,233,257,269,281,293,317,353,
%T A040117 389,401,449,461,509,521,557,569,593,617,641,653,677,701,761,773,797,
%U A040117 809,821,857,881,929,941,953,977,1013,1049,1061,1097,1109,1181,1193
%N A040117 Primes congruent to 5 (mod 12). Also primes p such that x^4 = 9 has no solution mod p.
%C A040117 Odd primes of the form 2x^2-2xy+5y^2 with x and y nonnegative. - _T. D. Noe_, May 08 2005, corrected by _M. F. Hasler_, Jul 03 2025
%C A040117 Complement of A040116 relative to A000040. - _Vincenzo Librandi_, Sep 17 2012
%C A040117 Odd primes of the form a^2 + b^2 such that a^2 == b^2 (mod 3). - _Thomas Ordowski_ and _Charles R Greathouse IV_, May 20 2015
%C A040117 _Yasutoshi Kohmoto_ observes that nextprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the next prime must be at a gap of 4 or 8 or 12..., but a gap of 4 is impossible because 12k + 5 + 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the next prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs 65% (as the above simple explanation suggests), but considering primes up to 10^8 yields a ratio of about 40% vs 60%. It can be expected that the ratio asymptotically tends to 1:1. - _M. F. Hasler_, Sep 01 2017
%H A040117 Vincenzo Librandi, <a href="/A040117/b040117.txt">Table of n, a(n) for n = 1..1000</a>
%F A040117 a(n) ~ 4n log n. - _Charles R Greathouse IV_, May 20 2015
%t A040117 Select[Prime/@Range[250], Mod[ #, 12]==5&]
%t A040117 ok[p_]:= Reduce[Mod[x^4 - 9, p] == 0, x, Integers] == False;Select[Prime[Range[200]], ok] (* _Vincenzo Librandi_, Sep 17 2012 *)
%o A040117 (PARI) for(i=1,250, if(prime(i)%12==5, print(prime(i))))
%o A040117 (Magma) [p: p in PrimesUpTo(1200) | not exists{x : x in ResidueClassRing(p) | x^4 eq 9} ]; // _Vincenzo Librandi_, Sep 17 2012
%Y A040117 Cf. A068227, A068228, A068229, A068231, A068232, A068233, A068234, A068235.
%Y A040117 Equal to A243183 (primes of the form 2x^2+2xy+5y^2) except for the additional A243183(1) = 2 (and indexing of subsequent terms).
%K A040117 nonn,easy
%O A040117 1,1
%A A040117 _N. J. A. Sloane_
%E A040117 More terms from _Dean Hickerson_, Feb 27 2002