Some factors of the numbers $G_{n}=6^{2n}+1$ and $H_{n}=10^{2n}+1$

Abstract:

All numbers $G_n = 6^{2n} + 1$ and $H_n = 10^{2n} + 1$ are searched for factors of the form $p = u \cdot 2^8 + 1 < 3.88 \cdot 10^{11}$ for $s \geqq n + 1$, and odd $u$. The search limit for $u$ was 60000 for $G_n$, and 156250 for $H_n$. A number of factors are found in this range. The numbers $G_6$ and $H_6$, lacking small factors, are proved composite by calculating $5^{(G_6 - 1)/2} (\bmod G_6)$ and $3^{(H_6 - 1)/2} (\bmod H_6)$, the residues found being different from $\pm 1$. The smallest numbers $G_n$ and $H_n$ with unknown characters are $G_{11}$ and $H_{10}$.