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

1,1

Comments

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers with (m-1)^2+n^2, m^2+(n-1)^2, (m+1)^2+n^2 and m^2+(n+1)^2 all prime.

It is easy to prove that if m and n are positive integers with (m-1)^2+n^2, m^2+(n-1)^2, (m+1)^2+n^2 and m^2+(n+1)^2 all prime, then either m = n = 2 or m == n == 0 (mod 5).

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.

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A000040, A000290, A236193, A259492, A261339.

Sequence in context: A274962 A375077 A199948 * A281692 A225221 A347654

Adjacent sequences: A261379 A261380 A261381 * A261383 A261384 A261385

Keyword

nonn

Author

Zhi-Wei Sun, Aug 17 2015

Status

approved