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A371303
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Numbers k > 4 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every natural m > 0 with 2^(2^m) < k.
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0
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7, 9, 15, 27, 57, 63, 195, 267, 363, 405, 483, 603, 1197, 1233, 1443, 1737, 2715, 4257, 5403, 6117, 21855, 22287, 26817, 40755, 63777, 260007, 617253, 986733, 1151655, 1167837, 1174503, 1199373, 1331595, 3233307, 4128873, 4138707, 4609527, 5938107, 7203945, 7605213, 8379405, 8587545, 9596223
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OFFSET
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1,1
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COMMENTS
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It seems that there are infinitely many such numbers.
If k > 7 is such a number, then it is odd and divisible by 3.
Conjecture: numbers k > 2 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every integer m >= 0 with 2^(2^m) < k are only 9, 15, and 195 (Amiram Eldar checked that there are no more terms k < 10^8).
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LINKS
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MATHEMATICA
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q[k_] := Module[{m = 1}, While[2^(2^m) < k && PrimeQ[k - 2^(2^m)] && PrimeQ[k + 2^(2^m)], m++]; 2^(2^m) > k]; Select[Range[5, 10^6, 2], q] (* Amiram Eldar, Mar 18 2024 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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