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A261382
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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