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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Conjecture about Proving Primality of Fermat numbers by Elliptic Curves technic
In 2008 and 2009, Denomme-Savin and Tsumura provided 2 papers providing a Primality Test for Fermat numbers based on Elliptic Curves technic:
$$ \text{Let } DST(x)= \frac{\displaystyle x^4+2x^2+1}{\...
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Smallest prime of the form $\frac{a^n + 1}{a+1}$ has $ 1 < n < a + 2$?
Consider primes of the form
$$\frac{a^n + 1}{a+1}$$
for integer $a>1$ and integer $n>1$.
Conjecture :
(for any fixed $a$)
The smallest prime of the form $\frac{a^n + 1}{a+1}$ has $ 1 < n < ...
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Help me correct my thinking of Fermat's last theorem
Fermat's last theorem as per the wiki states that
No three positive integers a, b, and c satisfy the equation $a^n + b^n = c^n$ for any integer value of $n>2$.
I just recently came across the ...
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Is the "reverse" of the $33$ rd Fermat number composite?
If we write down the digits of the $33$ rd Fermat number $$F_{33}=2^{2^{33}}+1$$ in base $10$ in reverse order , the resulting number should , considering its magnitude , be composite.
But can we ...
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Smallest prime factor of this $40$ million digit number?
Concatenate the Fermat numbers $F_0=2^{2^0}+1$ to $F_{26}=2^{2^{26}}+1$ in base $10$.
This gives $$3517257655374294967297\cdots9215379822913519617$$ a huge number with $$40\ 403\ 579$$ digits. Because ...
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Can you prove that my candidate PRP test for Wagstaff numbers (based on Elliptic Curve Primality Proving for Fermat numbers) is a true Primality Test?
The test is explained and described in the forum of the GIMPS project:
https://www.mersenneforum.org/showthread.php?t=28658
In short:
$$x_1=W_3=3 \ , \ x_{j+1} = \frac{x_j^4+2x_j^2+1}{4(x_j^3-x_j)}$$...
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for what all n is $2^n≡ 3\pmod{ 13}$ true. Please find below my initial steps [duplicate]
$a_n=(2^n)−3 $
, how do we find which $a_n$ are divisible by 13? Or we can re write it as
for what all n is $2^n≡ 3\pmod{13}$ true.
$16≡ 3mod 13$ or $2^4≡ 3\pmod{13}$
$2^{4n}≡ 3^{n}\pmod{13}$ or
$2^{...
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Fermat's Figurate Number Theory [closed]
$$\sum_{i=1}^n \frac{i(i+1)...(i+k-1)}{k!}=\frac{n(n+1)...(n+k)}{(k+1)!}$$
Regard "k" as a fixed positive integer,
How can I prove this formula of Fermat by using induction on "n". ...
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Is it possible that only finitely many Fermat numbers are composite?
The current status of the Fermat numbers $$F_n=2^{2^n}+1$$ where $n$ is a nonnegative integer , is that it is prime for $n\le 4$ and composite for $5\le n\le 32$
It is conjectured that only finitely ...
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Quadratic residues and their using in proving the Édouard Lucas theorem related to Fermat numbers. [duplicate]
I was doing a study on Fermat Numbers when I came across this theorem by Édouard Lucas (unproven in my reference material):
Every prime divisor of $F_n = 2^{2^{n}} + 1$ is of the form $k \cdot{2^{n+2}...
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Why can't I fit certain numbers back into the formula from Fermat's little theorem?
I'm given that the formula for Fermat's little theorem:
$$ a^{-1} \equiv a^{p-2} \ (mod \ p) $$
I presume I would then be able to get an inverse mod directly from this formula. Let's say $p=7$ and $a=...
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Proving that each Fermat number is prime iff $F_n | 3^{\frac{F_n-1}{2}}+1$.
I am getting into a bit of number theory and came up with the following exercise:
Exercise. Show that each Fermat number is prime iff $F_n | 3^{\frac{F_n-1}{2}}+1$.
My only idea follows (it is not a ...
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Is this proof, about Fermat numbers being coprime, wrong?
Let $m, n$ be positive integers, with $m > n$ and $2^{2^m}+1$ and $2^{2^n}+1$, Fermat numbers.
To prove that both Fermat numbers are coprime, it's sufficient to state $$\begin{align}(2^{2^m}+1, 2^{...
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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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