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
division carrying no additional information.
We compare the generalization obtained from Eulers factorization with
the generalization of the fragmentary factorization of Broda
using~$2^{640}\equiv 1 \pmod{641}$.
We show that this generalization can be used to
obtain factors of additional Fermat pseudo primes.
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