A105282 Positive integers n such that n^20 + 1 is semiprime (A001358).

2, 4, 46, 154, 266, 472, 748, 1434, 1738, 2058, 2204, 2222, 2428, 2478, 2510, 2866, 3132, 3288, 3576, 3688, 3756, 4142, 4506, 4940, 5164, 6252, 6330, 6786, 7180, 7300, 7338, 7416, 7628, 7806, 9270, 9312, 10044, 10722, 10860, 12126, 12422, 12668, 12998, 13350

Offset: 1

Author

Jonathan Vos Post, Apr 25 2005

Keywords

• easy
• nonn

Comments

We have the polynomial factorization: n^20 + 1 = (n^4 + 1) * (n^16 - n^12 + n^8 - n^4 + 1). Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^4+1 is prime and (n^16 - n^12 + n^8 - n^4 + 1) is prime.

Examples

			2^20 + 1 = 1048577 = 17 * 61681,
4^20 + 1 = 1099511627777 = 257 * 4278255361,
46^20 + 1 = 1799519816997495209117766334283777 = 4477457 * 401906666439788301510827761,
1434^20 + 1 =
1352019721694375552250489804528860551814233886722212960509362177 =
4228599998737 * 319732233386510278346888399489424537759394853595121.
		

Links

• Robert Price, Table of n, a(n) for n = 1..1405

Crossrefs

Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657.

Programs

Magma

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1000] | IsSemiprime(n^20+1)] // Vincenzo Librandi, Dec 21 2010

Mathematica

Select[Range[1000000], PrimeQ[#^4 + 1] && PrimeQ[(#^20 + 1)/(#^4 + 1)] &] (* Robert Price, Mar 09 2015 *)

Formula

a(n)^20 + 1 is semiprime (A001358).

Extensions

a(9)-a(44) from Robert Price, Mar 09 2015