Intereting Posts

Continuous increasing function with different Dini derivatives at 0
About Riemann's Hypothesis.
Constructing a family of distinct curves with identical area and perimeter
Two circumcircles of triangles defined relative to a fixed acute triangle are tangent to each other (IMO 2015)
Calculate Determinant A size n
First-order formula in first-order language, another open language where equivalence true on the naturals?
Infinite Product computation
Links between difference and differential equations?
Infinite subset of Denumerable set is denumerable?
Game Theory Optimal Solution to 2 Player Betting Game
Let $G$ be a graph of minimum degree $k>1$. Show that $G$ has a cycle of length at least $k+1$
Prove that $\sum\limits_{n=0}^{\infty}\frac{F_{n}}{2^{n}}= \sum\limits_{n=0}^{\infty}\frac{1}{2^{n}}$
Prove that limit goes to inf
Collection of numbers always in increasing or decreasing order
$\aleph_1 = 2^{\aleph_0}$

I tried to use quadratic reciprocity, but I don’t understand how to use the explicit formulae to end the problem. I surmised that the primes that work are congruent to $\pm1\bmod5$, but I could not generalise. where $F_{p-1}$ is the p-1th term of the Fibonacci Sequence

- If all of the integers from $1$ to $99999$ are written down in a list, how many zeros will have been used?
- Relationship between prime factorizations of $n$ and $n+1$?
- Translation of a certain proof of $(\sum k)^2 = \sum k^3 $
- When is a sum of consecutive squares equal to a square?
- If $x\equiv 1 \mod 3$, then $x^{100}\equiv 1 \mod 3$.
- Find all integer solutions to $7595x + 1023y=124$
- Count ways to take pots
- Proof that no polynomial with integer coefficients can only produce primes
- Polynomial $p(a) = 1$, why does it have at most 2 integer roots?
- Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Given the explicit formula

$$ F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right], $$

as soon as $p>5$ is a prime for which $5$ is a quadratic residue (i.e. a prime $p\equiv \pm 1\pmod{5}$ by quadratic reciprocity) we have

$$ F_{p-1}\equiv 0\pmod{p} $$

as a consequence of Fermat’s little theorem.

$\left(\frac{5}{p}\right)=+1$ allows us to consider $\sqrt{5}$ as an element of $\mathbb{F}_p^*$.

In general, $\sqrt{5}$ is an element of $\mathbb{F}_{p^2}$, hence $p\mid F_{p^2-1}$ for any prime $p\neq 5$.

Additionally, if $p\neq 5$ does not divide $F_{p-1}$, it divides $F_{p+1}$.

- Why is every discrete subgroup of a Hausdorff group closed?
- Visualising extra dimensions
- Prerequisite reading before studying the Collatz $3x+1$ Problem
- Taylor series expansion for $f(x)=\sqrt{x}$ for $a=1$
- Special linear group as a submanifold of $M(n, \mathbb R)$
- If $(A-B)^2=O_2$ then $\det(A^2 – B^2)=(\det(A) – \det(B))^2$
- Fixed Point of $x_{n+1}=i^{x_n}$
- Derivative with respect to $y'$ in the Euler-Lagrange differential equation
- Evaluate the integration : $\int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx$
- Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
- How to visualize a rank-2 tensor?
- How prove this inequality
- Rubik Cube finite non-abelian group
- unorthodox solution of a special case of the master theorem
- There is an element of order $51$ in the multiplicative group $(Z/103Z)^∗.$