# 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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### 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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### Fermat's little theorem, Poulet numbers, Carmichael numbers, and primes

Fermat's primality test for base 2 permits Poulet numbers to pass the test, as follows: $(2^x - 2)/x$. Fermat's primality test in different bases will act as a sieve for eliminating most pseudo ...
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### Prove that $\gcd(2^{2^{22}}+1,2^{2^{222}}+1)=1$

The great common divisor (gcd) of $2^{2^{22}}+1$ and $2^{{2}^{222}}+1$ is My work, \begin{align} F_{n}-2&= 2^{2^{n}}+1-2 \\ &=(2^{2^{n-1}}+1)(2^{2^{n-2}}+1)(2^{2^{n-2}}-1)\\ &=(2^{2^{m}}+1)... 1 vote
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### The greatest prime factor of $2^n+1$

Let $f(n)$ be the greatest prime factor of $2^n+1$ Is it true that for any $c>0$ ,there is an integer $n>c$ such that $f(n+1)<f(n)$ ? This is true if there are infinitely many Fermat primes:...
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### Fermat numbers, GCD and proving existence of infinite primes [closed]

Prove that if a, m, n are positive integers with $m ≠ n$, then $gcd({a^2}^n+1,{a^2}^m+1)=1$ if $a$ is even and $2$ if a is odd. Use this to show that there are infinitely many primes. WHAT?! i do not ...
1 vote
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### Show that $2^{(p-1)/2} \equiv 1\: \mathrm{mod}\:p$? [duplicate]

Let $F_n = 2^{2^n}+1$ be the $n$th Fermat number for $n \geq 2$ and $p$ a prime factor of $F_n$. How can I show that that $2^{(p-1)/2} \equiv 1 \:\mathrm{mod}\:p$?
### $F_n$ is the $n$-th Fermat Number. Prove there are infinitely many values of n for which $F_n + 2$ is composite.
$F_n$ is the $n$-th Fermat Number. $F_n = 2^{2^n} + 1$. Prove there are infinitely many values of n for which $F_n + 2$ is composite. I tried using reduce modulo 7 but got stuck. Any help is ...