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A105078
Positive integers n such that n^10 + 1 is semiprime.
13
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
OFFSET
1,1
COMMENTS
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.
LINKS
EXAMPLE
4^10+1 = 1048577 = 17 * 61681,
16^10+1 = 1099511627777 = 257 * 4278255361,
1010^10+1 = 1104622125411204510010000000001 = 1020101 * 1082855644108970101989901.
MATHEMATICA
Select[ Range[5000], PrimeQ[ #^2 + 1] && PrimeQ[(#^10 + 1)/(#^2 + 1)] &] (* Robert G. Wilson v, Apr 08 2005 *)
Select[Range[4700], PrimeOmega[#^10+1]==2&] (* Harvey P. Dale, Jan 13 2013 *)
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 06 2005
EXTENSIONS
More terms from Robert G. Wilson v, Apr 08 2005
STATUS
approved