Solution of Ljunggren Equation

Theorem

The only solutions of the Ljunggren equation:

$x^2 + 1 = 2 y^4$

are:

$x = 1, y = 1$

$x = 239, y = 13$

This sequence is A229384 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Setting $x = 1$:

\(\ds \)

\(\)

\(\ds 1^2 + 1\)

\(\ds \)

\(=\)

\(\ds 2\)

\(\ds \)

\(=\)

\(\ds 2 \times 1^4\)

and so $y = 1$.

Setting $x = 239$:

\(\ds \)

\(\)

\(\ds 239^2 + 1\)

\(\ds \)

\(=\)

\(\ds 57122\)

\(\ds \)

\(=\)

\(\ds 2 \times 13^4\)

and so $y = 13$.

This theorem requires a proof. In particular: It remains to be shown this is the only solution. Perhaps trial and error for $y$ going from $0$ up to $13$ and then using Largest Prime Factor of $n^2 + 1$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Largest Prime Factor of $n^2 + 1$

Sources

• 1942: W. Ljunggren: Zur Theorie der Gleichung $x^2 + 1 = D y^4$ (Avh. Norske Vid. Akad.Oslo.I. Vol. 27, no. 5: p. 27)

• 1991: Ray Steiner and Nikos Tzanakis: Simplifying the solution of Ljunggren's equation $X^2 + 1 = 2 Y^4$ (Journal of Number Theory Vol. 37, no. 2: pp. 123 – 132)

• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $239$