78,557 is Sierpiński

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Theorem

$78 \, 557$ is a Sierpiński number of the second kind.


Proof

When considering $\bmod {36}$, every positive integer $n$ can be written in one of the forms:

$2 k, 4 k + 1, 3 k + 1, 12 k + 11, 18 k + 15, 36 k + 27, 9 k + 3$

$\begin{array}{|c|c|} \hline n \bmod {36} & \text { Form } \\ \hline 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34 & 2 k \\ \hline 1, 5, 9, 13, 17, 21, 25, 29, 33 & 4 k + 1 \\ \hline 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34 & 3 k + 1 \\ \hline 11, 23, 35 & 12 k + 11 \\ \hline 15, 33 & 18 k + 15 \\ \hline 27 & 36 k + 27 \\ \hline 3, 12, 21, 30 & 9 k + 3\\ \hline \end{array}$

As seen by inspection, every number less than $36$ is included in the table.


And we have:

\(\displaystyle 78557 \times 2^{2 k} + 1\) \(\equiv\) \(\displaystyle 78557 \times 1^k + 1\) \(\displaystyle \pmod 3\) Fermat's Little Theorem and Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 2 + 1\) \(\displaystyle \pmod 3\) Congruence of Product
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod 3\)
\(\displaystyle 78557 \times 2^{4 k + 1} + 1\) \(\equiv\) \(\displaystyle 78557 \times 1^k \times 2 + 1\) \(\displaystyle \pmod 5\) Fermat's Little Theorem and Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 2 \times 2 + 1\) \(\displaystyle \pmod 5\) Congruence of Product
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod 5\)
\(\displaystyle 78557 \times 2^{3 k + 1} + 1\) \(=\) \(\displaystyle 78557 \times 8^k \times 2 + 1\)
\(\displaystyle \) \(\equiv\) \(\displaystyle 78557 \times 1^k \times 2 + 1\) \(\displaystyle \pmod 7\) Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 3 \times 2 + 1\) \(\displaystyle \pmod 7\) Congruence of Product
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod 7\)
\(\displaystyle 78557 \times 2^{12 k + 11} + 1\) \(\equiv\) \(\displaystyle 78557 \times 1^k \times 2^{11} + 1\) \(\displaystyle \pmod {13}\) Fermat's Little Theorem and Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 11 \times 2048 + 1\) \(\displaystyle \pmod {13}\) Congruence of Product
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod {13}\)
\(\displaystyle 78557 \times 2^{18 k + 15} + 1\) \(\equiv\) \(\displaystyle 78557 \times 1 \times 2^{15} + 1\) \(\displaystyle \pmod {19}\) Fermat's Little Theorem and Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 11 \times 32768 + 1\) \(\displaystyle \pmod {19}\) Congruence of Product
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod {19}\)
\(\displaystyle 78557 \times 2^{36 k + 27} + 1\) \(\equiv\) \(\displaystyle 78557 \times 1^k \times 2^{27} + 1\) \(\displaystyle \pmod {37}\) Fermat's Little Theorem and Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 6 \times 512^3 + 1\) \(\displaystyle \pmod {37}\) Congruence of Product
\(\displaystyle \) \(\equiv\) \(\displaystyle 6 \times \paren {-6}^3 + 1\) \(\displaystyle \pmod {37}\) Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod {37}\)
\(\displaystyle 78557 \times 2^{9 k + 3} + 1\) \(\equiv\) \(\displaystyle 78557 \times 512^k \times 2^3 + 1\)
\(\displaystyle \) \(\equiv\) \(\displaystyle 78557 \times 1^k \times 8 + 1\) \(\displaystyle \pmod {73}\) Congruence of Powers
\(\displaystyle \) \(\equiv\) \(\displaystyle 9 \times 8 + 1\) \(\displaystyle \pmod {73}\) Congruence of Product
\(\displaystyle \) \(\equiv\) \(\displaystyle 0\) \(\displaystyle \pmod {73}\)

Therefore $78557 \times 2^n + 1$ is always divisible by one of $\set {3, 5, 7, 13, 19, 37, 73}$.

Hence they are composite for every $n$.

$\blacksquare$


Historical Note

The fact that 78,557 is Sierpiński was proved by John Lewis Selfridge in $1962$.

It is still an open question whether it is the smallest.