Intereting Posts

Inner Product Spaces over Finite Fields
Countable ordinals are embeddable in the rationals $\Bbb Q$ — proofs and their use of AC
Linear transformations map lines to lines
How to Visualize points on a high dimensional (>3) Manifold?
Differential of transposed matrices
Algorithm to write an element of $SL_2(\mathbb{Z})$ as a product of $S, T^n$
Kernel of composition of linear transformations
Number of Subspaces that contains other Space
Prove $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$ using geometric approach
Failure of differential notation
Bifurcation Example Using Newton's Method
Prove that $(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$
Returning Paths on Cubic Graphs
How to find an irrational number in this case?
Comparing the expected stopping times of two stochastically ordered random processes (Prove or give a counterexample for the two claims)

Not sure how to go about this. Law of quadratic reciprocity and Euler’s Criterion is recently learned material but I’m not sure how this applies.

- When is $\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod {2^n} \ne 0\;$?
- Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer
- Is it possible to solve $i^2+i+1\equiv 0\pmod{2^p-1}$ in general?
- Prove: For odd integers $a$ and $b$, the equation $x^2 + 2 a x + 2 b = 0$ has no integer or rational roots.
- Pure Mathematics proof for $(-a)b$ =$-(ab)$
- Sum of n consecutive numbers
- Using gauss's lemma to find $(\frac{n}{p})$ (Legendre Symbol)
- $x^2+y^3 = z^4$ for positive integers
- If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?
- Primes in $\lfloor a^{n} \rfloor$

Note that by a Legendre symbol calculation, we have $(3/83)=1$. Since $3$ is a quadratic residue of $83$, it follows that $3^{(83-1)/2}\equiv 1\pmod{83}$. So $83$ is a prime divisor of your number.

Alternately, you could use Euler’s Criterion to show that $3$ is a quadratic residue of $83$. That involves somewhat more calculation than the calculation of the Legendre symbol (but less theory).

One answer is 83, as found by Andre (sorry for the accent .. )

Here is a way to educatedly “guess” it: If $p = 41a+1$ is prime, then both $p$ and $3^{41}-1$ divide $3^{41a} -1= 3^{p-1}-1$. There is a chance that $p$ divides $3^{41}-1$. Since $a=1$ doesn’t work, the next to try is $a=2$. Then $(3^{41})^2 -1 = (3^{41}-1)(3^{41}+1)$ is a multiple of 83. As $3^4 \equiv -2 \pmod{83}$, it follows that $3^{40} \equiv 1024 \equiv 28 \pmod{83}$, hence $3^{41} \equiv 1 \pmod{83}$.

Remark:

*Mathematica* claims that the prime factorization of $3^{41}-1$ is $2\cdot 83\cdot 2526913\cdot 86950696619$. All three odd primes are $\equiv 1\pmod{41}$

- Are Euclidean domains exactly the ones which we can define “mod” on?
- Prove or disprove: $f^{-1}(f(f^{-1}(Y))) = f^{-1}(Y)$.
- Normal Operators: Transform
- Evaluating $\int \cos^4(x)\operatorname d\!x$
- If $G$ is a groupe such that $|G|=p^m k$, does $G$ has a subgroup of order $p^n$ with $n<m$.
- To compute $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2|^2 dz$ where $\mathcal{C}$ is the unit circle in $\mathbb{C}$
- Why square matrix with zero determinant have non trivial solution
- Is infinity larger than 1?
- Maximize the value of $v^{T}Av$
- Selection of $b_n$ in Limit Comparison Test for checking convergence of a series
- If $X,Y$ are equivalence relations, so is $X \times Y$
- $\mathbb{Z}/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
- Expressing Factorials with Binomial Coefficients
- when the sum of some fractions be $1$
- Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?