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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 23 2003
STATUS
approved