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A105237
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Positive integers n such that n^13 + 1 is semiprime.
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13
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2, 22, 108, 126, 180, 256, 336, 490, 630, 652, 660, 682, 708, 760, 828, 862, 882, 1030, 1038, 1128, 1162, 1216, 1318, 1450, 1612, 1930, 1950, 2010, 2236, 2268, 2380, 2436, 2658, 2752, 2800, 2962, 2998, 3036, 3048, 3318, 3672, 3922, 4152, 4396, 4506, 4816
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OFFSET
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1,1
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COMMENTS
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We have the polynomial factorization: n^13+1 = (n+1) * (n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n+1 is prime and n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 is prime.
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LINKS
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EXAMPLE
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2^13+1 = 8193 = 3 * 2731,
22^13+1 = 282810057883082753 = 23 * 12296089473177511,
1030^13+1 = 1468533713451564313811276230000000000001 = 1031 * 1424377995588326201562828545101842871.
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MATHEMATICA
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Select[Range[0, 300000], PrimeQ[# + 1] && PrimeQ[(#^13 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)
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PROG
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(Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1600]|IsSemiprime(n^13+1)] // Vincenzo Librandi, Dec 21 2010
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CROSSREFS
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Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
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KEYWORD
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easy,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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