# On the generalization of factorizations of Fermat pseudo primes

## Matthias Baaz

L.~Euler proved by the following famous calculation that $F_5 = 4294967297$ is compound.
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\begin{eqnarray*}
5 \cdot 2^7 + 1 & \equiv & 0 \pmod{5 \cdot 2^7 + 1}\\
5 \cdot 2^7 & \equiv & -1 \pmod{5 \cdot 2^7 + 1}\\
5^4 \cdot 2^{7 \cdot 4} & \equiv & 1 \pmod{5 \cdot 2^7 + 1}\\
5^4 + 2^4 & = & 5 \cdot 2^7 +1\\
5^4 & \equiv & -2^4 \pmod{5 \cdot 2^7 + 1}\\
1 \equiv 5^4 \cdot 2^{7 \cdot 4} & \equiv & \underbrace{-2^4 \cdot
2^{7 \cdot 4}}_{-2^{2^5}} \pmod{\underbrace{(5 \cdot 2^7 + 1)}_{641}}
\end{eqnarray*}
The result can be derived immediately by direct division.
$$2^{2^5} + 1 = 641 \cdot 6700417$$
We presents a \emph{logical} method to extract the general content of
calculations.
This generalizations are applied to characterize the difference between the
mathematical insight that is used in the first factorization and the
using~$2^{640}\equiv 1 \pmod{641}$.