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Primes of the form a^2 + b^6.
3

%I #16 Dec 11 2020 11:29:55

%S 2,5,17,37,73,89,101,113,197,233,257,353,401,577,593,677,733,829,1129,

%T 1153,1213,1289,1297,1433,1601,1753,1913,2089,2273,2917,3089,3137,

%U 3229,3313,3433,4093,4177,4217,4289,4357,4457,4721,4937,5393,5477,5689,6121

%N Primes of the form a^2 + b^6.

%C Friedlander and Iwaniec prove that there are an infinite number of primes of the form a^2+b^4 (A028916). They speculate that the a^2+b^6 case can be proved by similar methods.

%H Vincenzo Librandi, <a href="/A078523/b078523.txt">Table of n, a(n) for n = 1..1000</a>

%H John Friedlander and Henryk Iwaniec, <a href="https://doi.org/10.1073/pnas.94.4.1054">Using a parity-sensitive sieve to count prime values of a polynomial</a>, PNAS February 18, 1997 94 (4) 1054-1058.

%H Jori Merikoski, <a href="https://arxiv.org/abs/2012.05675">A Cubic analogue of the Friedlander-Iwaniec spin over primes</a>, arXiv:2012.05675 [math.NT], 2020.

%e 73 = 3^2 + 2^6

%t maxN=10000; lst={}; Do[p=i^2+j^6; If[p<maxN&&PrimeQ[p], AppendTo[lst, p]], {i, maxN^(1/2)}, {j, maxN^(1/6)}]; lst=Union[lst]

%o (PARI) list(lim)=my(v=List([2]),b6,t); lim\=1; for(b=1,sqrtnint(lim-1,6), b6=b^6; forstep(a=1+b%2,sqrtint(lim-b6),2, if(isprime(t=a^2+b6), listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, Aug 18 2017

%Y Cf. A028916.

%K easy,nonn

%O 1,1

%A _T. D. Noe_, Nov 26 2002