A261382 Least positive integer k such that (k-1)^2+(k*n)^2, k^2+(k*n-1)^2, (k+1)^2+(k*n)^2 and k^2+(k*n+1)^2 are all prime.
2, 2510, 15, 30, 5, 510, 730, 440, 195, 6230, 2040, 2760, 20, 1010, 12570, 31340, 1625, 1650, 725, 2480, 2160, 520, 1055, 60, 5, 20, 1260, 25800, 6185, 6240, 10, 1180, 12600, 7500, 5330, 390, 325, 2880, 11655, 32670, 5850, 43110, 3230, 1470, 7680, 4950, 255, 202650, 10530, 450, 2445, 11670, 8745, 103350, 80, 6890, 135, 18930, 80, 245040
Offset: 1
Keywords
Examples
a(2) = 2510 since (2510-1)^2+(2510*2)^2 = 31495481, 2510^2+(2510*2-1)^2 = 31490461, (2510+1)^2+(2510*2)^2 = 31505521 and 2510^2+(2510*2+1)^2 = 31510541 are all prime.
References
- Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
- Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..60
- Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
Programs
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Mathematica
PQ[p_]:=PrimeQ[p] q[m_,n_]:=PQ[(m-1)^2+n^2]&&PQ[m^2+(n-1)^2]&&PQ[(m+1)^2+n^2]&&PQ[m^2+(n+1)^2] Do[k=0;Label[bb];k=k+1;If[q[k,k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}] -
PARI
is_ok(k,n)=isprime((k-1)^2+(k*n)^2)&&isprime(k^2+(k*n-1)^2)&&isprime((k+1)^2+(k*n)^2)&&isprime(k^2+(k*n+1)^2) first(m)=my(v=vector(m),k=1);for(i=1,m,while(!is_ok(k,i),k++);v[i]=k;k++;);v; \\ Anders Hellström, Aug 17 2015
Comments