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A057683
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Numbers n such that n^2+n+1, n^3+n+1 and n^4+n+1 are all prime.
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0
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1, 2, 5, 6, 12, 69, 77, 131, 162, 426, 701, 792, 1221, 1494, 1644, 1665, 2129, 2429, 2696, 3459, 3557, 3771, 4350, 4367, 5250, 5670, 6627, 7059, 7514, 7929, 8064, 9177, 9689, 10307, 10431, 11424, 13296, 13299, 13545, 14154, 14286, 14306, 15137
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| After a(0) = 1, it is never the case that n^5 + n + 1 is prime. Proof: consider integers modulo 4, that is, as 4n+k. (4*n+k)^5 + (4*n+k) + 1 factors into irreducible components over Z. 1024n^5 + 1280k(n^4) + 640(k^2)(n^3) + 160(k^3) (n^2) + (20(k^4)+4)n + (k^5+k+1) = (16n^2 + 8kn + 4n + k^2 + k + 1) (64n^3 + 48k(n^2) - 16n^2 + 12(k^2)n - 8kn + k^3 - k^2 + 1). - Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 17 2007
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EXAMPLE
| 5 is included because 5^2+5+1=31, 5^3+5+1=131 and 5^4+5+1=631 are all prime.
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MATHEMATICA
| Select[Range[16000], And@@PrimeQ/@(Table[n^i+n+1, {i, 2, 4}]/.n->#)&] (* From Harvey P. Dale, Mar 28 2011 *)
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PROG
| (MAGMA)[n: n in [0..20000]|IsPrime(n^2+n+1) and IsPrime(n^3+n+1) and IsPrime(n^4+n+1)][From Vincenzo Librandi, Dec 20 2010]
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CROSSREFS
| Cf. A049407.
Sequence in context: A108365 A064765 A082552 * A069480 A100613 A070911
Adjacent sequences: A057680 A057681 A057682 * A057684 A057685 A057686
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Harvey P. Dale (hpd1(AT)is2.nyu.edu), Oct 20 2000
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