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

1,1

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

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.

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Primes in A166687.

Cf. A079544, A079739, A079740.

Sequence in context: A147813 A338578 A274386 * A154755 A040095 A040028

Adjacent sequences: A079542 A079543 A079544 * A079546 A079547 A079548

Keyword

nonn

Author

N. J. A. Sloane, Jan 23 2003

Status

approved