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Search: id:A105078
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| A105078 |
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Positive integers n such that n^10 + 1 is semiprime. |
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+0
10
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| 4, 16, 26, 54, 110, 120, 126, 260, 314, 420, 444, 470, 570, 646, 714, 890, 946, 1010, 1294, 1306, 1394, 1640, 1674, 1794, 1920, 1964, 2116, 2174, 2360, 2430, 2624, 2666, 2884, 2924, 3094, 3106, 3174, 3220, 3504, 3686, 3826, 3884, 3924, 4046, 4540, 4700
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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We have the polynomial factorization: n^10+1 = (n^2+1) * (n^8 - n^6 + n^4 - n^2 + 1) Hence after the initial n=1 prime the binomial can only be semiprime if n^2 + 1 is prime and (n^8 - n^6 + n^4 - n^2 + 1) is prime.
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EXAMPLE
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4^10+1 = 1048577 = 17 * 61681,
16^10+1 = 1099511627777 = 257 * 4278255361,
1010^10+1 = 1104622125411204510010000000001 = 1020101 * 1082855644108970101989901.
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MATHEMATICA
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Select[ Range[5000], PrimeQ[ #^2 + 1] && PrimeQ[(#^10 + 1)/(#^2 + 1)] &] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 08 2005)
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CROSSREFS
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Cf. A000040, A001538, A103854, A104238, A105041, A105066.
Sequence in context: A163095 A075576 A111350 this_sequence A050707 A046346 A134330
Adjacent sequences: A105075 A105076 A105077 this_sequence A105079 A105080 A105081
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 06 2005
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 08 2005
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