Intereting Posts

Probability of drawing cards in ascending order
Ramsey Number proof: $R(3,3,3,3)\leq 4(R(3,3,3)-1) + 2$
$(P(X), C_R)$ may be a choice structure even if $R$ is not a rational relation.
Given a joint characteristic function, find $P(X<Y)$
Is there an efficient method to decide whether $gnu(n)<n$ , $gnu(n)=n$ or $gnu(n)>n$?
How to find the $E$ and $Var$ of an Itô Diffusion
Fourier transform of Schrödinger kernel: how to compute it?
Proving the inverse (if any) of a lower triangular matrix is lower triangular
Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
How to prove this $\pi$ formula?
$\lim_{n\rightarrow \infty } (\frac{(n+1)^{2n^2+2n+1}}{(n+2)^{(n+1)^2}\cdot n^{n^2}})$
How to prove whether a polynomial function is even or odd
Minimum number of out-shuffles required to get back to the start in a pack of $2n$ cards?
Is there a constructible flat pairing function?
Using de Moivre's Theorem to derive the relation…

Denote an artificial square E as a number:

$$E \in \Bbb{N}| \lnot (\exists y \in \Bbb{Z} | y^2 = E) \land (For \ each \ w \in \Bbb{Z} \ \exists a_w | a_w^2 \equiv E \ \pmod w) $$

In other words this numbers are able to pass every square test via modular arithmetic, but aren’t squares themselves.

- Showing there exists infinite $n$ such that $n! + 1$ is divisible by atleast two distinct primes
- If $(a^{n}+n ) \mid (b^{n}+n)$ for all $n$, then $ a=b$
- Algorithms for Finding the Prime Factorization of an Integer
- Integer $2 \times 2$ matrices such that $A^n = I$
- For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$
- A type of integer numbers set

My guess is they don’t exist. Simply because for a sufficiently large w. It will be clear that no number squares to E but I’m not sure if this is rigorous enough of an argument, or if I have somehow forgotten detail

- If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer
- Doubts about a nested exponents modulo n (homework)
- Determine if a number is the sum of two triangular numbers.
- Does someone know why raising the element of a group to the power of the order of the group yields the identity?
- Diophantine equation involving factorial …
- Show that $p$ is prime if the following limit property holds
- Prerequisites for Dirichlet & Dedekind's Vorlesungen über Zahlentheorie
- Proof that $a\mid b \land b\mid c \Rightarrow a\mid c $
- Multiplicative property of the GCD
- Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime

Suppose that $E = p^{2k+1} n$ where $(n,p) = 1$. Take $w = p^{2k+2}$. If $E$ is a quadratic residue modulo $p^{2k+2}$ then there exists $m$ such that

$$ p^{2k+2} \mid p^{2k+1} n – m^2. $$

In particular, $p^{2k+1} \mid p^{2k+1} n – m^2$ and so $p^{2k+1} \mid m^2$. Since $m$ is a (bone fide) square, in fact $p^{2k+2} \mid m^2$, and so $p^{2k+2} \mid p^{2k+1} n$. But that implies $p \mid n$, contradicting our initial assumption.

- Strange set notation (a set as a power of 2)?
- metric on the Euclidean Group
- Why does Monte-Carlo integration work better than naive numerical integration in high dimensions?
- Limit of a particular variety of infinite product/series
- Multiplicative group of integers modulo n definition issues
- sum of series using mean value theorem
- Prove that a non-positive definite matrix has real and positive eigenvalues
- A formula for the power sums: $1^n+2^n+\dotsc +k^n=\,$?
- Proof that every natural number is the sum of 9 cubes of natural numbers
- On convergence of nets in a topological space
- Can we expect to find some constant $C$; so that, $\sum_{n\in \mathbb Z} \frac{1}{1+(n-y)^{2}} <C$ for all $y\in \mathbb R;$?
- Orientation preserving diffeomorphism.
- Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
- Factorial and exponential dual identities
- Intuition behind the Axiom of Choice