%I #41 Mar 26 2022 00:36:59
%S 2,3,5,11,17,19,37,41,53,59,73,83,101,107,131,137,149,163,179,181,197,
%T 227,233,251,257,293,307,347,389,401,443,467,491,521,523,563,577,587,
%U 593,613,641,677,739,773,809,811,821,883
%N Primes of the form x^2 + y^2 + 1 with x,y >= 0.
%C Bredihin proves that this sequence is infinite. Motohashi improves the upper and lower bounds. - _Charles R Greathouse IV_, Sep 16 2011
%C Sun & Pan prove that there are arbitrarily long arithmetic progressions in this sequence. - _Charles R Greathouse IV_, Mar 03 2018
%C For this sequence in short intervals, see Wu and Matomäki; for its Goldbach problem, see Teräväinen. - _Charles R Greathouse IV_, Oct 10 2018
%H Charles R Greathouse IV, <a href="/A079545/b079545.txt">Table of n, a(n) for n = 1..10000</a>
%H B. M. Bredihin, <a href="http://mi.mathnet.ru/eng/izv3124">Binary additive problems of indeterminate type II. Analogue of the problem of Hardy and Littlewood</a> (in Russian). Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 27 (1963), pp. 577-612.
%H M. N. Huxley and H. Iwaniec, <a href="https://doi.org/10.1112/S0025579300006069">Bombieri's theorem in short intervals</a>, Mathematika 22 (1975), pp. 188-194.
%H Henryk Iwaniec, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21118.pdf">Primes of the type φ(x, y) + A where φ is a quadratic form</a>, Acta Arithmetica 21 (1972), pp. 203-234.
%H Kaisa Matomäki, <a href="http://users.utu.fi/ksmato/papers/Primesm2n2p1.pdf">Prime numbers of the form p = m^2 + n^2 + 1 in short intervals</a>, Acta Arithmetica 128 (2007), pp. 193-200.
%H Y. Motohashi, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1642.pdf">On the distribution of prime numbers which are of the form x^2 + y^2 + 1</a>. Acta Arithmetica 16 (1969), pp. 351-364.
%H Y. Motohashi, <a href="https://doi.org/10.1007/BF01896011">On the distribution of prime numbers which are of the form x^2 + y^2 + 1. II"</a>, Acta Mathematica Academiae Scientiarum Hungaricae 22 (1971), pp. 207-210.
%H Yu-Chen Sun and Hao Pan, <a href="https://arxiv.org/abs/1708.08629">The Green-Tao theorem for primes of the form x^2 + y^2 + 1</a>, arXiv:1708.08629 [math.NT], 2017.
%H Joni Teräväinen, <a href="https://arxiv.org/abs/1611.08585">The Goldbach problem for primes that are sums of two squares plus one</a>, Mathematika 64 (2018), pp. 20-70. arXiv:1611.08585 [math.NT], 2016-2017.
%H J. Wu, <a href="https://doi.org/10.1090/S0002-9939-98-04414-1">Primes of the form p = 1 + m^2 + n^2 in short intervals</a>, Proceedings of the American Mathematical Society 126 (1998), pp. 1-8.
%F Iwaniec proves that a(n) ≍ n (log n)^(3/2), that is, n (log n)^(3/2) << a(n) << n (log n)^(3/2). - _Charles R Greathouse IV_, Mar 06 2018
%e 17 = 0^2 + 4^2 + 1 is prime so in this sequence.
%t Select[Select[Range[1000], SquaresR[2, #] != 0&]+1, PrimeQ] (* _Jean-François Alcover_, Aug 31 2018 *)
%o (PARI) list(lim)={
%o my(A,t,v=List([2]));
%o forstep(a=2,sqrt(lim-1),2,
%o A=a^2+1;
%o forstep(b=0,min(a,sqrt(lim-A)),2,
%o if(isprime(t=A+b^2),listput(v,t))
%o )
%o );
%o forstep(a=1,sqrt(lim-2),2,
%o A=a^2+1;
%o forstep(b=1,min(a,sqrt(lim-A)),2,
%o if(isprime(t=A+b^2),listput(v,t))
%o )
%o );
%o vecsort(Vec(v),,8)
%o }; \\ _Charles R Greathouse IV_, Sep 16 2011
%o (PARI) is(n)=for(x=sqrtint(n\2),sqrtint(n-1), if(issquare(n-x^2-1), return(isprime(n)))); 0 \\ _Charles R Greathouse IV_, Jun 12 2015
%o (PARI) B=bnfinit('x^2+1);
%o is(n)=!!#bnfisintnorm(B,n-1) && isprime(n) \\ _Charles R Greathouse IV_, Jun 13 2015
%Y Primes in A166687.
%Y Cf. A079544, A079739, A079740.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jan 23 2003
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