Intereting Posts

Proving that $\sqrt{2}+\sqrt{3}$ is irrational
Approximating a piecewise continuous function with a function in $\mathcal{C}^{\infty}_{0}(\mathbb{R})$
Constructor theory distinguishability
If $f$ is a real analytic funtion, then the solutions of $\dot{x} = f(x)$ are analytic as well
Question about Grassmannian, most vectors in $\bigwedge^k V$ are not completely decomposable?
Has knot theory led to the development of better knots?
Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?
a geometry problem about inscribed and circumscribed circle radius.
When finding root, does Newton's method fail if the function is non-differentiable?
Uncountability of countable ordinals
Gödel, Escher, Bach: $ b $ is a power of $ 10 $.
“sheaf” au sens de Serre
Basis of a projection
Extensions: Spectrum
Calculate volume of intersection of 2 spheres

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.

- Proving $x^4+y^4=z^2$ has no integer solutions
- How to find the last digit of $3^{1000}$?
- Show that $\pi =4-\sum_{n=1}^{\infty }\frac{(n)!(n-1)!}{(2n+1)!}2^{n+1}$
- Expressing an integer as the sum of three squares
- If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
- Parametrization of solutions of diophantine equation $x^2 + y^2 = z^2 + w^2$
- How to determine whether a number can be written as a sum of two squares?
- Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$.
- Find all $x,y\in \mathbb{N}$ such that: $2^x+17=y^2$.
- numbers' pattern

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 $\equiv 1\pmod{41}$

- Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
- Probability and Data Integrity
- Calculating residue of pole of order $2$
- existence of the solution of Neumann problem in $\mathbb{R}^3$
- Tensors: Acting on Vectors vs Multilinear Maps
- Finding a space $X$ such that $\dim C(X)=n$.
- Find for which value of the parameter $k$ a function is bijective
- Derivative of $\frac{1}{\sqrt{x+5}}$
- Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$
- Show that the polynomial $x^3-xy^2+y+1$ is irreducible in $\mathbb{Q}$
- All ideals of a subring of $\Bbb Q$
- Is this an inner product on $L^1$?
- convergence of a tower power
- Is it valid to write $1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$?
- Find a non-zero integer matrix $X$ such that $XA=0$ where $X,A,0$ are all $4 \times 4$