Intereting Posts

What is an example of real application of cubic equations?
Pullback of maximal ideal in $k$ is not maximal in $k$.
Combinatorial Proof Of ${n \choose k}={n-1\choose {k-1}}+{n-1\choose k}$
For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
Show that $A∩B∩C= ∅$ is only true when $A∩B = ∅, A∩C = ∅$ or $B∩C = ∅$ or show a counterexample.
How can we show that $\mathbb Q$ is not free?
What are the surfaces of constant Gaussian curvature $K > 0$?
2 color theorem
What kind of vector spaces have exactly one basis?
Graph type taxonomy
What are the most overpowered theorems in mathematics?
Factorial sum estimate $\sum_{n=m+1}^\infty \frac{1}{n!} \le \frac{1}{m\cdot m!}$
Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure
concepts which is present in metric space but not in topological space
Challenge: How to prove this identity between bi- and trinomial coefficients?

Prove that if $n \in \mathbb{N}$, then $$\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$$ where $d(n)$ is the divisor function. I have know only information is $d(n)=\sum_{d|n}{1}$.

- Maps of primitive vectors and Conway's river, has anyone built this in SAGE?
- Does Chaitin's constant have infinitely many prime prefixes?
- Problems about consecutive semiprimes
- Integer parts of multiples of irrationals
- Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers
- Irrationality/Transcendentality of values of $e^{e^x}$
- Chicken Mcnugget Theorem (Frobenous Coin) Problem
- Can the sum of the first $n$ squares be a cube?
- Number of ordered pairs of coprime integers from $1$ to $N$
- The n-th prime is less than $n^2$?

**Outline:** Let $f(n)$ be the function on the left, and $g(n)$ the function on the right.

Both $f$ and $g$ are **multiplicative**.

So we only need to verify that the equality holds when $n$ is a power of a prime.

Multiplicativity then atutomatically gives us the rest.

The number $p^k$ has $k+1$ divisors: in symbols, $d(p^k)=k+1$.

So we want to prove that for any $m$,

$$\sum_0^m (k+1)^3 =\left(\sum_0^m (k+1)\right)^2.$$

Look up the formula for the sum of the first $q$ cubes.

- Asymptotic difference between a function and its binomial average
- Could someone explain the concept of a set being “open relative” to another set?
- If $f$ and $g$ are integrable, then $h(x,y)=f(x)g(y)$ is integrable with respect to product measure.
- What's the correct notation for log squared?
- Tautological line bundle over rational projective space
- Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.
- Galois groups of polynomials and explicit equations for the roots
- Proving $\sqrt{ab} = \sqrt a\sqrt b$
- Construct series
- Can we expect to find some constant $C$; so that, $\sum_{n\in \mathbb Z} \frac{1}{1+(n-y)^{2}} <C$ for all $y\in \mathbb R;$?
- Prove $\int_{0}^{\infty}{\ln x\ln\left(x\over 1+x\right)\over (1+x)^2}dx$ and $\int_{0}^{\infty}{\ln x\ln\left(x\over 1+x\right)\over 1+x}dx$
- Let $K$ be a field and $f(x)\in K$. Prove that $K/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K$.
- Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$
- Evaluate $\lim_{x\to \infty}\ (x!)^{1/x}$
- Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$