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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: $$3,\ 5,\ 17,\ 257,\ 65537,\ \cdots $$

If $2^k +1$ is prime, and $k > 0$, it can be shown that $k$ must be a power of two, $k=2^n$. A number of the form $2^{2^n}+1$ is called a Fermat number, and when it happens to be a prime, it is called a Fermat prime. As of 2017, the only known Fermat primes are those for which $0\leq n\leq 4$. In addition, John L. Selfridge made an intriguing conjecture: Let $g(n)$ be the number of distinct prime factors of $2^{2^n} + 1$. Then $g(n)$ is not monotonic (non-decreasing).

If another Fermat prime exists, that would imply that the conjecture is false.

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If $2^n+1$ is prime, why must $n$ be a power of $2$?

A little bird told me that if $2^n+1$ is prime, then $n$ is a power of $2$. I tend not to trust talking birds, so I'm trying to verify that statement independently. Suppose $n$ is not a power of $2$. Then $n = a \cdot 2^m$ for some $a$ not a power…
Rob
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Prime divisors of Fermat numbers $2^{2^n}+1$ have form $k\cdot 2^{n+2}+1$

A theorem of Édouard Lucas related to the Fermat numbers states that : Prime divisors of Fermat numbers $2^{2^n}+1$ have form $k\cdot 2^{n+2}+1$ Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater…
Pedja
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If $b$ is even and not a power of two, can $b^4+1$ be a Fermat pseudoprime base 2?

The complete question is already in the title but we shall provide some motivation as well. We study generalized Fermat numbers defined by: $$\mathrm{GF}(n,b) = b^{2^n}+1$$ where $b$ and $n$ are natural numbers and we are interested in cases where…
Jeppe Stig Nielsen
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Can a composite number $3\cdot 2^n + 1$ divide a Fermat number $2^{2^m}+1$?

In OEIS entry A204620, there is a question (by Arkadiusz Wesolowski) about whether composite numbers of the form: $$3\cdot 2^n + 1$$ can be divisors of a Fermat number, i.e. of a number of the form $2^{2^m}+1$. The question is "answered" by a…
Jeppe Stig Nielsen
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To show that Fermat number $F_{5}$ is divisible by $641$.

How can I show that Fermat number $F_{5}=2^{2^5}+1$ is divisible by $641$.
Kns
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How to prove that if a prime $p$ divides a Fermat Number then $p=k\cdot 2^{n+2}+1$?

If a prime $p$ divides a Fermat Number then $p=k\cdot 2^{n+2}+1$? Does anyone know a simple/elementary proof?
Pierre Fermat
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Is my proof correct on how $k$ must be a power of $2$? Are there other proofs?

So I was looking at the Fermat Primes. These are primes of the form $2^k+1$ for a natural number $k$, such that I define by $\mathbb{N}:=\big\{1,2,3,\ldots\big\}$ and $0\notin \mathbb{N}$. We denote by $F_1$ the first Fermat Prime; $F_2$ the second…
Mr Pie
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Why are the first 5 Fermat numbers prime?

The $n$th Fermat number $F_n$ is defined as $F_n = 2^{2^n}+1$. The first five Fermat numbers, $F_0,F_1,F_2,F_3,F_4$, are all prime. Why is this? It seems like a fairly surprising coincidence that all five of these are prime. The prime number…
D.W.
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Use Fermat’s little theorem to show that $8, 9, 10$ are not prime numbers

Use Fermat’s little theorem to show that $8, 9, 10$ are not prime numbers. I know that the theorem states: for all $a$ in $\mathbb Z$, if $p$ is prime and $p$ does not divide $a$ then $a^p \equiv a$ mod $p$, which means that $a^{p-1} \equiv 1$ mod…
cbc bc
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Easiest way to check whether the number is prime or not

Recently I came across a YouTube video which explains the easiest way to check whether the given number is prime or not the equation was: $$\frac{2^x - 2}{x}$$ According to that video if $x$ is a prime number, it will give a whole number as a…
Gautam Krishna R
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Induction on Fermat Numbers: $F_n = \prod_{j=0}^{n-1}F_j+2$

Is the Following Proof Correct? Theorem. Given that $\forall n\in\mathbf{N}(F_n = 2^{2^n}+1)$ show that the following is true $$\forall n\in{1,2,3...}\left(F_n = \prod_{j=0}^{n-1}F_j+2\right)$$ Proof. We construct the proof by recourse to…
atifcppprogrammer
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What is the connection between Fermat's Little Theorem and "Fermat Liars"?

I know that Fermat's Little Theorem states that if $p$ is prime and $1 < a < p$, then $a^{p-1} \equiv 1 ($mod $p)$. I also know that a Fermat Liar is any $a$ such that $a^{n-1} \equiv 1 ($mod $n$), when $n$ is composite. I feel that these two points…
mathstack
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Show that every composite Fermat number is a pseudoprime base 2.

Question: Show that every composite Fermat number $F_m=2^{2^m}+1$ is a pseudoprime base 2. Hint: Raise the congruence $2^{2^m}\equiv-1($mod $F_m)$ to the $2^{2^m-m}$th power. Even with the hint, I'm fairly lost. I understand that…
Ephraim
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Is the "reverse" of the $33$ rd Fermat number composite?

If we write down the digits of the $33$ rd Fermat number $$F_{33}=2^{2^{33}}+1$$ in base $10$ in reverse order , the resulting number should , considering its magnitude , be composite. But can we search for a prime factor WITHOUT actually…
Peter
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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 that reads: "$F_0 $ and $F_1$ ($3$ and $5$ respectively) are…
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