Largest Prime Factor of n squared plus 1

Theorem

Let $n \in \Z$ be greater than $239$.

Then the largest prime factor of $n^2 + 1$ is at least $17$.

Proof

We note that for $n = 239$ we have:

\(\ds \)

\(\)

\(\ds 239^2 + 1\)

\(\ds \)

\(=\)

\(\ds 57122\)

\(\ds \)

\(=\)

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

Thus the largest prime factor of $239^2 + 1$ is $13$.

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Also see

• Solution of Diophantine Equation $x^2 + 1 = 2 y^4$

Sources

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

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