Intereting Posts

What are the theorems in mathematics which can be proved using completely different ideas?
weak sequential continuity of linear operators
Sum identity using Stirling numbers of the second kind
Bernoulli's representation of Euler's number, i.e $e=\lim \limits_{x\to \infty} \left(1+\frac{1}{x}\right)^x $
Fixed Point of $x_{n+1}=i^{x_n}$
Eigenvalues of matrix with entries that are continuous functions
What is the definition of a commutative diagram?
Bounds for $\zeta$ function on the $1$-line
Is there a geometric meaning to the outer product of two vectors?
Average norm of a N-dimensional vector given by a normal distribution
Minimal prime ideals of $\mathcal O_{X,x}$ correspond to irreducible components of $X$ containing $x$
Limit of nth root of n!
Expected Value Function
What is the limit of the rank of the power of a matrix?
Showing when a permutation matrix is diagonizable over $\mathbb R$ and over $\mathbb C$

Show that if $x, y, z$ are positive integers, then $(xy + 1)(yz + 1)(zx + 1)$ is a perfect square if and only if $xy + 1, yz + 1, zx+1$ are perfect squares.

- Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?
- Cover $\{1,2,…,100\}$ with minimum number of geometric progressions?
- Integer solutions to the equation $a^3+b^3+c^3=30$
- Do these inequalities regarding the gamma function and factorials work?
- combinatorial question (sum of numbers)
- How can $p^{q+1}+q^{p+1}$ be a perfect square?
- Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $$ for every natural number $m$?
- Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of prime numbers, then $\lim_{n \to \infty}\frac{p_{n+1}}{p_n} = 1$?
- Why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?
- Adding or Multiplying Transcendentals

Here’s the brief summary of the article posted by sdcvvc. It is essentially a proof by descent, showing that if you had a triple $(x,y,z)$ such that the product $(xy+1)(yz+1)(xz+1)$ was a square with one of the three factors *not* a square, then you could find a smaller such triple (ordered, say, via the sum $x+y+z$.)

The descent is rather direct: If $(x,y,z)$ is a triple with $x\leq y\leq z$, then so is $(x,y,z')$, where

$$

z'=x+y+z+2xyz-2\sqrt{(xy+1)(xz+1)(yz+1)}

$$

(Recall that the term under the square root was assumed square.) Their remains some checking to do; namely, that this is indeed such a triple, and that that $0<z'<z$, but this is all rather straight-forward.

Lest this seem entirely *ad hoc*, let me just note that, as I learned from Kedlaya’s article, that sets of this type (with the property that pairwise products are of a fixed distance from a square…in our case we are learning about sets $\{x,y,z\}$ with each pairwise product one less than a square) have been heavily studied by Fermat, Diophantus, and a slew of more modern mathematicians, featuring some applications of Baker’s theory of linear forms in logarithms. I’d recommend taking a look at the original article — it’s brief, informative, and well-written.

- Finite partition of a group by left cosets of subgroups
- what operation repeated $n$ times results in the addition operator?
- Prove that $\sum_{k=1}^n \frac{2k+1}{a_1+a_2+…+a_k}<4\sum_{k=1}^n\frac1{a_k}.$
- What are some examples of infinite dimensional vector spaces?
- How to Compute Genus
- Range of a standard brownian motion, using reflection principle
- Dedekind cuts for $\pi$ and $e$
- Is $ \pi $ definable in $(\Bbb R,0,1,+,×, <,\exp) $?
- What can I do with proper classes?
- Isomorphisms Between a Finite-Dimensional Vector Space and its Dual
- $\int \ln (\cos x)\,dx$
- Integral of bounded continuous function on $R$
- Prove or disprove: $(\mathbb{Q}, +)$ is isomorphic to $(\mathbb{Z} \times \mathbb{Z}, +)$?
- What is the transformation representation/interpretation of symmetric matrices?
- How to Make an Introductory Class in Set Theory and Logic Exciting