Intereting Posts

Proving that every element of a monoid occurs exactly once
Show that $y(t) = t$ and $g(t) = t \ln(t)$ are linearly independent
Dr Math and his family question. How to solve without trial and error?
Showing that a CCC with a zero object is the trivial category
Finite rings without zero divisors are division rings.
What's the intuition behind Pythagoras' theorem?
Advantage of ZF over other set theories such as New Foundation
Argument on why $\Bbb N$ does not have a least upper bound
How to compute $\limsup$ and $\liminf ,\;$ as $x\to+\infty,\;$ of $\;\sin(x^2+x+1/2)\sin(x+1/2)$?
Example of infinite field of characteristic $p\neq 0$
Limits problem with trig? Factoring $\cos (A+B)$?
How to show $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R}) \subset \mathcal{L}(\mathbb{R^2})$?
How to explain the formula for the sum of a geometric series without calculus?
How to write an integral as a limit?
Rearrange all the real numbers between $0$ and $1$

Let $k \geq 2$ be a positive integer and let $n=2^k+1$. How can I prove that $n$ is a prime number if and only if

$$3^{\frac{n-1}{2}} \equiv -1 \pmod n.$$

*Fixed.*

- Perhaps a Pell type equation
- Sum of digits of repunits
- Prove that $112$ divides the integral part of $4^m-(2+\sqrt{2})^m$
- If $n\ne 4$ is composite, then $n$ divides $(n-1)!$.
- Analyzing whether there is always a prime between $n^2$ and $n^2+n$
- Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$?

- Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$
- Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer
- Roots of Artin-Schreier equation
- Numbers as sum of distinct squares
- perfect square of the form $n^2+an+b$
- Tight bounds for Bowers array notation
- Factoring extremely large integers.
- If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
- Fibonacci modular results and $\, \gcd(F_n,F_m) = F_{\gcd(n,n)}$
- Cardinality of sum-set

Here are two options for finding a proper proof for this theorem (called Pepin’s test).

1) http://en.wikipedia.org/wiki/Pepin's_test.

2) “Solved and Unsolved Problems in Number Theory” by Daniel Shanks.

This book includes the proof for that theorem.

This is the simplest case of Pratt certificates for primality – have a look at http://mathworld.wolfram.com/PrattCertificate.html for a better explanation. (In the notation of the article, your question corresponds to the case where the only $p_i$ is $2$.)

- Can a rectangle be written as a finite almost disjoint union of squares?
- Cauchyness vs. Double Limits
- Representing the dirac distribution in $H^1(\mathbb R)$ through the scalar product
- Show that there exists a $3 × 3$ invertible matrix $M$ with entries in $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = I_3$.
- Explicit Riemann mappings
- Are all measure zero sets measurable?
- Finding out the area of a triangle if the coordinates of the three vertices are given
- Find closed formula for the recurrence $a_{n}=na_{n-1}+n(n-1)a_{n-2}$
- In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected
- The algebraic closure of a finite field and its Galois group
- $\cap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$
- The set of all infinite binary sequences
- Isomorphisms between the groups $U(10), U(5)$ and $\mathbb{Z}/4\mathbb{Z}$
- If $\sin\theta+\sin\phi=a$ and $\cos\theta+ \cos\phi=b$, then find $\tan \frac{\theta-\phi}2$.
- Inhomogeneous 2nd-order linear differential equation