A079545 Primes of the form x^2 + y^2 + 1 with x,y >= 0.

2, 3, 5, 11, 17, 19, 37, 41, 53, 59, 73, 83, 101, 107, 131, 137, 149, 163, 179, 181, 197, 227, 233, 251, 257, 293, 307, 347, 389, 401, 443, 467, 491, 521, 523, 563, 577, 587, 593, 613, 641, 677, 739, 773, 809, 811, 821, 883

Offset: 1

Author

N. J. A. Sloane, Jan 23 2003

Keywords

• nonn

Comments

Bredihin proves that this sequence is infinite. Motohashi improves the upper and lower bounds. - Charles R Greathouse IV, Sep 16 2011

Sun & Pan prove that there are arbitrarily long arithmetic progressions in this sequence. - Charles R Greathouse IV, Mar 03 2018

For this sequence in short intervals, see Wu and Matomäki; for its Goldbach problem, see Teräväinen. - Charles R Greathouse IV, Oct 10 2018

Examples

			17 = 0^2 + 4^2 + 1 is prime so in this sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• B. M. Bredihin, Binary additive problems of indeterminate type II. Analogue of the problem of Hardy and Littlewood (in Russian). Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 27 (1963), pp. 577-612.
• M. N. Huxley and H. Iwaniec, Bombieri's theorem in short intervals, Mathematika 22 (1975), pp. 188-194.
• Henryk Iwaniec, Primes of the type φ(x, y) + A where φ is a quadratic form, Acta Arithmetica 21 (1972), pp. 203-234.
• Kaisa Matomäki, Prime numbers of the form p = m^2 + n^2 + 1 in short intervals, Acta Arithmetica 128 (2007), pp. 193-200.
• Y. Motohashi, On the distribution of prime numbers which are of the form x^2 + y^2 + 1. Acta Arithmetica 16 (1969), pp. 351-364.
• Y. Motohashi, On the distribution of prime numbers which are of the form x^2 + y^2 + 1. II", Acta Mathematica Academiae Scientiarum Hungaricae 22 (1971), pp. 207-210.
• Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2 + y^2 + 1, arXiv:1708.08629 [math.NT], 2017.
• Joni Teräväinen, The Goldbach problem for primes that are sums of two squares plus one, Mathematika 64 (2018), pp. 20-70. arXiv:1611.08585 [math.NT], 2016-2017.
• J. Wu, Primes of the form p = 1 + m^2 + n^2 in short intervals, Proceedings of the American Mathematical Society 126 (1998), pp. 1-8.

Crossrefs

Primes in A166687.

Cf. A079544, A079739, A079740.

Programs

Mathematica

Select[Select[Range[1000], SquaresR[2, #] != 0&]+1, PrimeQ] (* Jean-François Alcover, Aug 31 2018 *)

PARI

list(lim)={
    my(A,t,v=List([2]));
    forstep(a=2,sqrt(lim-1),2,
        A=a^2+1;
        forstep(b=0,min(a,sqrt(lim-A)),2,
            if(isprime(t=A+b^2),listput(v,t))
        )
    );
    forstep(a=1,sqrt(lim-2),2,
        A=a^2+1;
        forstep(b=1,min(a,sqrt(lim-A)),2,
            if(isprime(t=A+b^2),listput(v,t))
        )
    );
    vecsort(Vec(v),,8)
}; \\ Charles R Greathouse IV, Sep 16 2011

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

PARI

B=bnfinit('x^2+1);
is(n)=!!#bnfisintnorm(B,n-1) && isprime(n) \\ Charles R Greathouse IV, Jun 13 2015

Formula

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