78,557 is Sierpiński
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:
| \(\ds 78557 \times 2^{2 k} + 1\) | \(\equiv\) | \(\ds 78557 \times 1^k + 1\) | \(\ds \pmod 3\) | Fermat's Little Theorem and Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 2 + 1\) | \(\ds \pmod 3\) | Congruence of Product | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod 3\) | |||||||||||
| \(\ds 78557 \times 2^{4 k + 1} + 1\) | \(\equiv\) | \(\ds 78557 \times 1^k \times 2 + 1\) | \(\ds \pmod 5\) | Fermat's Little Theorem and Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 2 \times 2 + 1\) | \(\ds \pmod 5\) | Congruence of Product | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod 5\) | |||||||||||
| \(\ds 78557 \times 2^{3 k + 1} + 1\) | \(=\) | \(\ds 78557 \times 8^k \times 2 + 1\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 78557 \times 1^k \times 2 + 1\) | \(\ds \pmod 7\) | Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 3 \times 2 + 1\) | \(\ds \pmod 7\) | Congruence of Product | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod 7\) | |||||||||||
| \(\ds 78557 \times 2^{12 k + 11} + 1\) | \(\equiv\) | \(\ds 78557 \times 1^k \times 2^{11} + 1\) | \(\ds \pmod {13}\) | Fermat's Little Theorem and Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 11 \times 2048 + 1\) | \(\ds \pmod {13}\) | Congruence of Product | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {13}\) | |||||||||||
| \(\ds 78557 \times 2^{18 k + 15} + 1\) | \(\equiv\) | \(\ds 78557 \times 1 \times 2^{15} + 1\) | \(\ds \pmod {19}\) | Fermat's Little Theorem and Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 11 \times 32768 + 1\) | \(\ds \pmod {19}\) | Congruence of Product | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {19}\) | |||||||||||
| \(\ds 78557 \times 2^{36 k + 27} + 1\) | \(\equiv\) | \(\ds 78557 \times 1^k \times 2^{27} + 1\) | \(\ds \pmod {37}\) | Fermat's Little Theorem and Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 6 \times 512^3 + 1\) | \(\ds \pmod {37}\) | Congruence of Product | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 6 \times \paren {-6}^3 + 1\) | \(\ds \pmod {37}\) | Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {37}\) | |||||||||||
| \(\ds 78557 \times 2^{9 k + 3} + 1\) | \(\equiv\) | \(\ds 78557 \times 512^k \times 2^3 + 1\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 78557 \times 1^k \times 8 + 1\) | \(\ds \pmod {73}\) | Congruence of Powers | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 9 \times 8 + 1\) | \(\ds \pmod {73}\) | Congruence of Product | ||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {73}\) |
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.