A079544 Primes of the form x^2 + y^2 + 1, x>0, y>0.
3, 11, 19, 41, 53, 59, 73, 83, 101, 107, 131, 137, 149, 163, 179, 181, 227, 233, 251, 293, 307, 347, 389, 401, 443, 467, 491, 521, 523, 563, 587, 593, 613, 641, 677, 739, 773, 809, 811, 821, 883
Offset: 1
Keywords
References
- D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- Aminu Alhaji Ibrahim, Sa’idu Isah Abubaka, Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties, Advances in Pure Mathematics, 2016, 6, 409-419.
- Ju. V. Linnik. An asymptotic formula in an additive problem of Hardy and Littlewood (Russian). Izv. Akad. Nauk SSSR, ser. math., 24:629-706, 1960. Cited in Matomäki 2007.
- Kaisa Matomäki, Prime numbers of the form p = m^2+n^2+1 in short intervals, Acta Arith. 128 (2007), pp. 193-200.
- Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2+y^2+1, arXiv preprint arXiv:1708.08629 [math.NT], 2017.
Crossrefs
Cf. A079545.
Programs
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Mathematica
iMax=7!; a=Floor[Sqrt[iMax]]; lst={}; Do[Do[p=x^2+y^2+1; If[PrimeQ@p&&p<=iMax,AppendTo[lst,p]],{y,1,a}],{x,1,a}]; Union[lst] (* Vladimir Joseph Stephan Orlovsky, Aug 11 2009 *) -
PARI
list(lim)=my(v=List(),t); lim\=1; for(x=1,sqrtint(lim-2), forstep(y=2-x%2,min(x,sqrtint(lim-x^2-1)), 2, if(isprime(t=x^2+y^2+1), listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 13 2012
Comments