A105934 Positive integers n such that n^22 + 1 is semiprime (A001358).

116, 176, 184, 300, 444, 470, 584, 690, 696, 950

Offset: 1

Author

Jonathan Vos Post, Apr 26 2005

Keywords

• easy
• nonn

Comments

We have the polynomial factorization: n^22 + 1 = (n^2 + 1) * (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1). Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^2+1 is prime and (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1) is prime.

Examples

			116^22 + 1 = 2618639792014920380336685706161496723088736257 = 13457 * 194593133091693570657404005808240820620401,
300^22 + 1 = 3138105960900000000000000000000000000000000000000000001 = 90001 * 34867456593815624270841435095165609271008099910001,
950^22 + 1 = 323533544973709366507562922501564025878906250000000000000000000001 = 902501 * 358485525194663902319845543109164450653136395416736380347501.
		

Crossrefs

Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282.

Programs

Magma

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..1000] | IsSemiprime(n^22+1)]; // Vincenzo Librandi, Dec 21 2010

Mathematica

Select[Range[1000], PrimeOmega[#^22 + 1]==2&] (* Vincenzo Librandi, May 24 2014 *)

Formula

a(n)^22 + 1 is in A001358. a(n)^2+1 is in A000040 and (a(n)^20 - a(n)^18 + a(n)^16 - a(n)^14 + a(n)^12 - a(n)^10 + a(n)^8 - a(n)^6 + a(n)^4 - a(n)^2 + 1) is in A000040.