A078900 Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>0.

5, 13, 17, 25, 41, 61, 85, 97, 113, 145, 181, 221, 257, 265, 313, 337, 365, 421, 481, 545, 613, 685, 761, 841, 881, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1921, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3697

Offset: 1

Author

T. D. Noe, Dec 12 2002

Keywords

• easy
• nonn

Comments

It can be shown that, like the Fermat numbers, two of these generalized Fermat numbers are coprime if they have the same base k. However, unlike the Fermat numbers (which are conjectured to be squarefree), these generalized Fermat numbers are not necessarily squarefree for k > 1. Riesel tabulates some prime factors of generalized Fermat numbers for k <= 5.

For k=1, these are the Fermat numbers A000215. See A078901 for the case m>1, which excludes the sum of consecutive squares. By Legendre's theorem (Riesel, p. 165), the prime factors of a generalized Fermat number are of the form 1 + f 2^(m+1) for some integer f.

References

• H. Riesel, "Prime numbers and computer methods for factorization," Second Edition, Progress in Mathematics, Vol. 126, Birkhauser, Boston, 1994, pp. 417-425.

Links

• T. D. Noe, Factorizations of Generalized Fermat Numbers
• Eric Weisstein's World of Mathematics, Generalized Fermat Number

Crossrefs

Cf. A000215, A078901.

Programs

Mathematica

mx=5000; maxK=Ceiling[Sqrt[mx/2]]; maxM=Ceiling[Log[2, Log[2, mx]]]; lst={}; Do[gf=(k+1)^2^m+k^2^m; If[gf
				

Extensions

Offset corrected by Sean A. Irvine, Jul 23 2025