Eisenstein's Conjecture

False Conjecture

Every natural number of the form:

$2^2 + 1, 2^{2^2} + 1, 2^{2^{2^2} } + 1, \ldots$

is prime.

Refutation

$2^{2^{16} } + 1$ is composite:

$2^{2^{2^{2^2} } } + 1 = 2^{2^{16} } + 1 = 825 \, 753 \, 601 \times 188 \, 981 \, 757 \, 975 \, 021 \, 318 \, 420 \, 037 \, 633 \times c$

where $c$ is a composite number with $19 \, 694$ digits.

$\blacksquare$

Source of Name

This entry was named for Ferdinand Gotthold Max Eisenstein.

Historical Note

Eisenstein's Conjecture was proven false in $1953$, when John Lewis Selfridge discovered the factor $825 \, 753 \, 601$ using the SWAC.

Sources

• 1953: J.L. Selfridge: Note $156$ -- Factors of Fermat numbers (MTAC Vol. 7: pp. 274 – 275)  www.jstor.org/stable/2002843

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Eisenstein, Ferdinand Gotthold Max (1823-52)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Eisenstein, Ferdinand Gotthold Max (1823-52)