A104238 Positive integers n such that n^5 + 1 is semiprime.

2, 10, 12, 16, 22, 126, 136, 180, 256, 268, 276, 366, 388, 396, 438, 462, 606, 642, 652, 658, 676, 738, 760, 768, 982, 1012, 1068, 1116, 1230, 1276, 1320, 1452, 1488, 1530, 1618, 1692, 1698, 1752, 1846, 1948, 1996, 2080, 2112, 2160, 2332, 2392, 2440, 2520

Offset: 1

Author

Jonathan Vos Post, Apr 02 2005

Keywords

• easy
• nonn

Comments

n^5+1 can only be prime when n = 1, n^5+1 = 2. This is because of the polynomial factorization n^5+1 = (n+1) * (n^4 - n^3 + n^2 - n + 1) = (n+1)*A060884(n). Hence after the initial n=1 prime, the binomial can at best be semiprime and that only when both (n+1) and (n^4 - n^3 + n^2 - n + 1) are primes.

Examples

			n n^5+1 = (n+1) * (n^4 - n^3 + n^2 - n + 1)
2 33 = 3 * 11
10 100001 = 11 * 9091
12 248833 = 13 * 19141
16 1048577 = 17 * 61681
		

Links

• Robert Price, Table of n, a(n) for n = 1..1414

Crossrefs

Cf. A001358, A103854.

Programs

Mathematica

Select[Range[2600],PrimeOmega[#^5+1]==2&] (* Harvey P. Dale, May 20 2011 *)
Select[Range[200000], PrimeQ[# + 1] && PrimeQ[(#^5 + 1)/(# + 1)] &] (* Robert Price, Mar 09 2015 *)

Formula

a(n)^5 + 1 is semiprime. a(n)+1 is prime and a(n)^4 - a(n)^3 + a(n)^2 - a(n) + 1 is prime.