

A040117


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


29



5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, 521, 557, 569, 593, 617, 641, 653, 677, 701, 761, 773, 797, 809, 821, 857, 881, 929, 941, 953, 977, 1013, 1049, 1061, 1097, 1109, 1181, 1193
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OFFSET

1,1


COMMENTS

Primes of the form 2x^22xy+5y^2 with x and y nonnegative.  T. D. Noe, May 08 2005.
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


LINKS



FORMULA



MATHEMATICA

Select[Prime/@Range[250], Mod[ #, 12]==5&]
ok[p_]:= Reduce[Mod[x^4  9, p] == 0, x, Integers] == False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 17 2012 *)


PROG

(PARI) for(i=1, 250, if(prime(i)%12==5, print(prime(i))))
(Magma) [p: p in PrimesUpTo(1200)  not exists{x : x in ResidueClassRing(p)  x^4 eq 9} ]; // Vincenzo Librandi, Sep 17 2012


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



