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A261339
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Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.
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2
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1, 1, 47500, 20440, 2, 124560, 17850, 2730, 185550, 1, 518910, 429180, 10, 687480, 81030, 36, 1568340, 2, 1165750, 7410, 10, 6780, 481140, 10, 10, 5430, 240, 2730, 72660, 2080, 18700, 291720, 295080, 52860, 5430, 1, 81030, 56400, 12490, 43590, 124560, 40030, 5170, 278700, 2091850, 131320, 184110, 11206510, 12910, 1245780
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OFFSET
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1,3
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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 with m and n in the set {k>0: k+1, k^2+1 and k^2+prime(k)^2 are all prime}.
For example, 5/8 = 3567600/5708160 with 3567600+1, 3567600^2+1 = 12727769760001, 3567600^2 + prime(3567600)^2 = 3567600^2 + 60098671^2 = 3624578025726241, 5708160+1, 5708160^2+1 = 32583090585601, and 5708160^2 + prime(5708160)^2 = 5708160^2 + 99018553^2 = 9837256928799409 all prime.
The conjecture implies that there are infinitely many primes p with (p-1)^2+1 and (p-1)^2+prime(p-1)^2 both prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with m^2+prime(m)^2, m^2+prime(n)^2, n^2+prime(m)^2 and n^2+prime(n)^2 all prime.
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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(3) = 47500 since 47501, 47500^2 + 1 = 2256250001, 47500^2 + prime(47500)^2 = 47500^2 + 578827^2 = 337296945929, 47500*3 + 1 = 142501, (47500*3)^2 + 1 = 20306250001, and (47500*3)^2 + prime(47500*3)^2 = 142500^2 + 1907023^2 = 3657042972529 are all prime.
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MATHEMATICA
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PQ[n_]:=PrimeQ[n+1]&&PrimeQ[n^2+1]&&PrimeQ[n^2+Prime[n]^2]
Do[k=0; Label[bb]; k=k+1; If[PQ[k]&&PQ[k*n], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]
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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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