No Fermat pseudoprime of the form u²−2 to base 2

THE PRIME NUMBER SERIES 31

No Fermat pseudoprime of the form to base

**Theorem.**
There exists no composite odd number

$U=u^2-2 \qquad (u\ge3\ \text{ungerade}),$

for which

$2^{U-1}\equiv1\pmod U$

holds.

—

**Proof.**
We carry out the proof by contradiction.

Assume that there exists a composite odd number

$U=u^2-2$

with

$2^{U-1}\equiv1\pmod U.$

Then is a Fermat pseudoprime to base .
For every prime divisor we have

(1) $\operatorname{ord}_p(2)\mid (U-1)=u^2-3.$

From $p\mid u^2-2$ follows

$u^2\equiv2\pmod p,$

thus $2$ thus is a quadratic residue modulo $p$ . Thus it holds

$\left(\frac{2}{p}\right)=1,$

and by the law of quadratic reciprocity

(2) $p\equiv\pm1\pmod8.$

Since is odd, we have $u^2\equiv1\pmod8$ , therefore

$p\equiv u^2-2\equiv7\pmod8.$

In particular $(p-1)/2$ is odd.
Because $2$ is a square modulo $p$ , follows

(3) $\operatorname{ord}_p(2)\mid\frac{p-1}{2}.$

From (1) and (3) we obtain

(4) $\operatorname{ord}_p(2)\mid \gcd\!\left(u^2-3,\frac{p-1}{2}\right).$

**Lemma.** For every prime divisor $p\mid(u^2-2)$ we have

$\gcd\!\left(u^2-3,\frac{p-1}{2}\right)=1.$

*Proof of the Lemma.*
From $p\mid u^2-2$ exists a $k\in\mathbb{Z}$ with

$u^2=2+kp,$

thus

$u^2-3=kp-1.$

Let

$d=\gcd\!\left(u^2-3,\frac{p-1}{2}\right).$

Then we have

$d\mid(kp-1),\qquad d\mid\frac{p-1}{2}.$

From $d\mid\frac{p-1}{2}$ follows

(5) $p\equiv1\pmod{2d}.$

Reduction of $u^2=2+kp$ modulo $d$ yields, because of (5)

(6) $u^2\equiv2+k\pmod d.$

On the other hand, it follows from $d\mid(u^2-3)$

(7) $u^2\equiv3\pmod d.$

From (6) and (7) follows

$k\equiv1\pmod d.$

Thus it holds

$kp-1\equiv p-1\pmod d.$

Because $d\mid(kp-1)$ , follows $d\mid(p-1)$ .
Because $(p-1)/2$ is odd, it follows

$d\mid\gcd(p-1,\tfrac{p-1}{2})=1,$

thus $d=1$ .

—

From (4) and the lemma it follows

$\operatorname{ord}_p(2)=1,$

thus

$2\equiv1\pmod p,$

which is impossible for a prime number .

The contradiction shows that the assumption was false.
Thus there exists no composite odd number of the form

$U=u^2-2,$

that is a Fermat pseudoprime to base 2.

Munich, 11 January 2026

Gottfried Färberböck

Archived version:
G. Färberböck, “No Fermat pseudoprime of the form u²−2 to base 2”,
Zenodo (2026).
https://doi.org/10.5281/zenodo.18232110