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My best conjectures

If $n>1$ is an odd natural number, then there are natural numbers $a,b$ such that $n=a+b$ and $a^2+b^2\in\mathbb P$.

Let $S_n=\Sigma^n_{k=1}p_k$, where $p_k$ is the $k$-th prime number. $\forall p\in\mathbb P\exists n\in\mathbb N: p|S_n$

$x\ge 13\implies\pi(x)\ge \pi\circ\pi(x)+\pi\circ\pi\circ\pi(x)+\cdots$.

For $N=\prod p_k^{n_k}>49$ it holds that $\prod k^{n_k}\leq\pi(N)$.

For all sets of non constant polynomials $\{f_1,\dots,f_n\}\subset \mathbb Z[X]$, it exists $m\in\mathbb N$ such that $f_k(m)\notin\mathbb P$, for all $1\leq k\leq n$. Where $\mathbb P$ is the set of primes.

For all $(m,n)\in\mathbb Z_+^2$ except $(3,4),(4,3) \text{ and } (4,4)$ it holds that $p_m\cdot p_n > p_{m\cdot n}$.

For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$.

Any even number $n\ge 36$ can be written as $n=a+b+c+d$ where $a^2+b^2+c^2=d^2$ and $a,b,c,d\in\mathbb Z^+$.

For each prime $p$ there are an infinite number of primes $q$ such that $p+q$ is a perfect square.

Any integer $n>462$ can be written as $n=ab+ac+bc$, where $a,b,c\in\mathbb Z_+$.

Given $a,b\in\mathbb Z^+$, and let $F_{a,b}:\mathbb N\to\mathbb N$ be a function such that $F_{a,b}(0)=0$ and $F_{a,b}(n+1)=a\cdot F_{a,b}(n)+b\cdot F_{a,b}(n-1)$. If $a,b$ are co-prime and $F_{a,b}(1)\in \mathbb Z^+$, then $F_{a,b}(\gcd(m,n))=\gcd(F_{a,b}(m),F_{a,b}(n))$.

A condition on strictly increasing sequences of natural numbers $(a_n)_n$: $\bigg\lfloor\frac{a_n^2}{a_{n+1}}\bigg\rfloor=2a_n-a_{n+1}\tag 1$ that (when $n$ is big enough) seems to hold for all increasing sequences $a_n\lesssim p_n$, where $p_n$ is the $n$-th prime.

All even numbers $n>2$ can be written $n=a+b$ where $a,b\in\mathbb N^+$ and $\frac{a^2+b^2}{2}\in\mathbb P$.

There is a real number $\gamma>1$ such that every odd number greater than $1$ can be written in the form $p+2\lfloor a^\gamma\rfloor$. Some $p\in\mathbb P$ and $a\in\mathbb N$.

Let $q_n$ be the $n$-th natural number that can be divided by a square $>1$. $\sum_{k=1}^\infty\frac{(-1)^{1+\Omega(q_k)}}{q_k}=0$ where $\Omega(n)$ is the number of (not necessarily different) prime factors of $n$.

Given two topological spaces $\left\langle X,\tau\right\rangle$, $\left\langle Y,\sigma\right\rangle$ and a function $X\overset{f}\longrightarrow Y$. Then it holds for all sets $M\subset Y$ that $x\in \overline{f^{-1}(M)}\Rightarrow f(x)\in \overline{M}$ iff $f$ is continuous.

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