Intereting Posts

$f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.
A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x^{(n)} dx$
Serre duality as a right adjoint functor
No consistent theory can define a product of distributions: why?
Why do the first spikes in these plots point in opposite directions?
Is axiom of choice necessary for proving that every infinite set has a countably infinite subset?
$\lim_{n\rightarrow \infty } (\frac{(n+1)^{2n^2+2n+1}}{(n+2)^{(n+1)^2}\cdot n^{n^2}})$
surface area of torus of revolution
Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.
Find all subrings of $\mathbb{Z}^2$
Algebraic values of sine at sevenths of the circle
Maximising sum of sine/cosine functions
Calippo, toothpaste and milk .. packing
Proof of Existence of Algebraic Closure: Too simple to be true?
Existence of sequences converging to $\sup S$ and $\inf S$

Question:If $n$ is a nonnegative integer, prove that $n + 2$ and $n^2 + n + 1$ cannot both be perfect cubes.

Possible solution:Suppose $n+2$ and $n^2 + n + 1$ are perfect cubes, their product $(n+2)(n^2 + n + 1)$ must also be a perfect cube.However, note that $(n+2)(n^2 + n + 1)=n^3 + 3n^2 + 3n + 2 = (n + 1)^3 + 1^3$

- Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
- If every $0$ digit in the expansion of $\sqrt{2}$ in base $10$ is replaced with $1$, is the resulting sequence eventually periodic?
- Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$
- regularization of sum $n \ln(n)$
- Prime factorization knowing n and Euler's function
- Why don't we define division by zero as an arbritrary constant such as $j$?
By Fermat’s Last Theorem, $a^n + b^n \neq c^n $ if $a,b,c,n$ are positive integers and $n>2$, therefore $a^3 + b^3 \neq c^3$ and $(n + 1)^3 + 1^3$ cannot be a perfect cube (can’t be expressed in the form $c^3$ where $c$ is a positive integer)

I’m looking for alternative methods of solution, and some verification that the above proof is correct.

- Show that if $c\mid a-b$ and $c\mid a'-b'$ then $c\mid aa'-bb'$
- Solve $x^3=y^2-7$?
- $p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
- Find all primes $p$ such that $\dfrac{(2^{p-1}-1)}{p}$ is a perfect square
- Can a Mersenne number ever be a Carmichael number?
- a conjectured continued-fraction for $\displaystyle\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$
- Asymptotic formula for almost primes
- Numbers of the form $\frac{xyz}{x+y+z}$, second question
- A convergence problem： splitting a double sum
- Calculating 7^7^7^7^7^7^7 mod 100

Except for $-1$, $0$ and $1$, the distance between consecutive perfect cubes is always greater than one. This is enough to conclude that $(n+1)^3+1$ is not a perfect cube when $n$ is nonnegative. (No need to invoke Fermat.)

- Proving $d$ is a metric of a power set
- Is there a memorable solution to Kirkman's School Girl Problem?
- A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$
- Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
- Proof by Induction: $2(\sqrt{n+1} – \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
- Closed-form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$
- Construct a function that takes any value even number of times.
- Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
- Realization of Bessel functions
- Why do bell curves appear everywhere?
- Limit of a recursive sequence $s_n = (1-\frac{1}{4n^2})s_{n-1}$
- Intuitive explation for oriented matroids?
- Does convergence in probability preserve the weak inequality?
- Is this *really* a categorical approach to *integration*?
- Find 3rd largest number out of 7 under at most 11 comparisons