# Questions tagged [fermat-numbers]

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is an integer of the form $$F_{n} = 2^{2^n} + 1$$ where $n$ is a nonnegative integer. The first few Fermat numbers are: $$3,\ 5,\ 17,\ 257,\ 65537,\ \cdots$$

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### Why does the Carlyle circle method fail to produce a regular n-gon for non-prime Fermat numbers?

The Carlyle circle method readily produces a regular pentagon, 17-gon, 257-gon and it seems the 65537-gon DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon ...
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### Help identify this prime number theorem [duplicate]

Studying thoroughly a physics matrix system I found (I'm pretty sure that I'm not committing errors) that for $P$ prime number and $n$ any integer $n<P$, we can decompose, $$\binom{P}{n}=P \ q$$ ...
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### Find solution using infinite descent.

Can someone help with a task? Need to find a solution other than $(0,0,0)$ with infinite descent. $x,y,z\in\mathbb{Z}$. Any help would be appreciated. The equation is $x^2-3y^2=2z^2$. I tried to ...
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### proof that $(b^n-1)/(b-1)$ is not prime if n is a pseudoprime not prime of the base $b$.

The question of my exercise says: Proof that, if $n$ is a pseudoprime not prime of the base $b$ (i.e. $b^{n-1}\equiv 1 (\mod n)$) then $N=(b^n-1)/(b-1)$ is also a pseudoprime not prime. I have proven ...
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### Tweaking Fermat's primality test - Why does it work and should I expect different results?

Fermat's primality test states that for any $p$ as a positive odd integer, IF: $$a^{{p-1}}\equiv 1{\pmod {p}}$$ or in a different way: $$a^{{p}}-a\equiv 0{\pmod {p}}$$ THEN $p$ is probably prime. ...
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### Prove that a squared number is an equivalence relation of $-1\pmod{p}$ [duplicate]

Prove that a squared number is an equivalence relation of $-1\pmod{p}$. Lets assume that $p$ is a prime number which satisfies: $$p \,\equiv\, 1 \pmod{4}.$$ How can one find a natural number $n$, ...
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### Is $3$ the only prime that is both a Mersenne prime and a Fermat prime?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
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### Prove that no Fermat number is a $3$ rd power of an integer.

Let $F_n$ be the $n$ th Fermat number, $F_n:=2^{2^n}+1$. I have been working with a similar question that reads: "Prove that no Fermat number is a perfect square." Where I found an answer ...
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