A105282 - OEIS

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

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OFFSET

1,1

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.

LINKS

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

FORMULA

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

EXAMPLE

2^20 + 1 = 1048577 = 17 * 61681,

4^20 + 1 = 1099511627777 = 257 * 4278255361,

46^20 + 1 = 1799519816997495209117766334283777 = 4477457 * 401906666439788301510827761,

1434^20 + 1 =

1352019721694375552250489804528860551814233886722212960509362177 =

4228599998737 * 319732233386510278346888399489424537759394853595121.

MATHEMATICA