Intereting Posts

Conditions for Schur decomposition and its generalization
Continuous uniform distribution over a circle with radius R
Diagonalize the matrix A or explain why it can't be diagonalized
Show that the area of a triangle is given by this determinant
Software for drawing and analyzing a graph?
Dimension of the sum of subspaces
Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc
Integral extension (Exercise 4.9, M. Reid, Undergraduate Commutative Algebra)
A derivation of the Euler-Maclaurin formula?
categorical interpretation of quantification
Presentation of $D_{2n}$
Parseval's Identity (Integral)
When adding zero really counts …
Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?
Integral Representation of Infinite series

Triangular numbers (See https://en.wikipedia.org/wiki/Triangular_number )

are numbers of the form $$\frac{n(n+1)}{2}$$

In ProofWiki I found three claims about triangular numbers. The three claims are that a triangular number cannot be a cube, not a fourth power and not a fifth power. Unfortunately, neither was a proof given nor did I manage to do it myself. Therefore my qeustions :

- Number of solutions of $x_1+2x_2+\cdots+kx_k=n$?
- $p$ be a prime number , then is it true that $p^2| {pa \choose pb}-{a \choose b} , \forall a,b \in \mathbb N$ ?
- Are there any Combinatoric proofs of Bertrand's postulate?
- Natual density inside a subsequence
- Cramér's Model - “The Prime Numbers and Their Distribution” - Part 1
- What is the *middle* digit of $3^{100000}$?

Does someone know a proof that a triangular number cannot be a cube, a fourth power or a fifth power ?

- Every Real number is expressible in terms of differences of two transcendentals
- Largest prime below a given number N
- Is there a power of 2 that, written backward, is a power of 5?
- $\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_k}$ not an integer
- Solve $x^3 +1 = 2y^3$
- Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$
- Finite sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$
- Consecutive Prime Factors
- Understanding a proof of the Davenport-Rado-Mirsky-Newman theorem
- Are fractions with prime numerator and denominator dense?

This problem was finished off (for arbitrary powers) in a paper of Gyory in 1997 (Acta Arithmetica) :

http://matwbn.icm.edu.pl/ksiazki/aa/aa80/aa8038.pdf

There are no unexpected solutions. The proof appeals to Darmon and Merel’s result on the equation $x^n+y^n=2z^n$ (though, with some care, it should be possible nowadays to prove it using only linear forms in logarithms).

First, notice $n$ and $n+1$ are coprime. And if the product of coprime numbers is a n-th power then both are also n-th powers. Now divide the problem into the cases where $n$ is odd and even.

$$n=2t$$

$$t(2t+1)=a^b$$

Then $t$ and $2t+1$ are b-th powers. Let $t=y^b$, $2t+1=x^b$. Then

$$x^b-2y^b=1$$

Applying the same substitutions to the case where $n$ is odd you find $$x^b-2y^b=-1$$

In this answer Keith Conrad proves the only solution is $x=1$, $y=0$, which mean $n=0$.

- Infinite partition of $\mathbb N$ by infinite subsets
- A good book for learning mathematical trickery
- Getting the shortest paths for chess pieces on n*m board
- If $a$ is a quadratic residue modulo every prime $p$, it is a square – without using quadratic reciprocity.
- An open interval is an open set?
- Are axioms assumed to be true in a formal system?
- find Limit $a_n= \frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{n+n}$
- Maximize and Minimize a 12″ piece of wire into a square and circle
- Every manifold admits a vector field with only finitely many zeros
- If a lottery has 300 tickets, shouldn't I win every 300 times I play
- Finding number of matrices whose square is the identity matrix
- Is there a change of variables that allows one to calculate $\int_0^\pi \frac{1}{4-3\cos^2 x}\, \mathrm dx$ avoiding improperties?
- Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$
- Proving Holder's inequality using Jensen's inequality
- Question on showing a bijection between $\pi_1(X,x_0)$ and $$ when X is path connected.