Open Problem Garden

Number Theory » Computational N.T.

Magic square of squares ★★

Author(s): LaBar

Question Does there exist a $3\times 3$ magic square composed of distinct perfect squares?

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Posted by maxal
updated November 23rd, 2008

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Diophantine quintuple conjecture ★★

Author(s):

Definition A set of m positive integers $\{a_1, a_2, \dots, a_m\}$ is called a Diophantine $m$ -tuple if $a_i\cdot a_j + 1$ is a perfect square for all $1 \leq i < j \leq m$ .

Conjecture (1) Diophantine quintuple does not exist.

It would follow from the following stronger conjecture [Da]:

Conjecture (2) If $\{a, b, c, d\}$ is a Diophantine quadruple and $d > \max \{a, b, c\}$ , then $d = a + b + c + 2bc + 2\sqrt{(ab+1)(ac+1)(bc+1)}.$

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Posted by maxal
updated February 25th, 2020

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Inverse Galois Problem ★★★★

Author(s): Hilbert

Conjecture Every finite group is the Galois group of some finite algebraic extension of $\mathbb Q$ .

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Posted by tchow
updated October 13th, 2008

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Gersgorin theorem states that: all the eigenvalues of $A=[a_{ij}]\in M_n$ are located in the union of $n$ discs $\bigcup\limits_{i=1}^n\{z\in C:|z-a_{ii}|\leq \sum\limits_{j=1,j\neq i}^n|a_{ij}|\}$ . For some special matrices, the region can be confined to $\bigcup\limits_{i=1}^n\{z\in C:|z-a_{ii}|\leq \sum\limits_{j=1,j\neq i}^n|a_{ij}|\}\backslash\{z\in C:|z-a_{kk}|<\sum\limits_{j=1,j\neq k}^n|a_{kj}|\}$ for some $k$ . I wonder if the new region above is valid in general?

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Posted by Miwa Lin
updated November 18th, 2009

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Algebra

trace inequality ★★

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Let $A,B$ be positive semidefinite, by Jensen's inequality, it is easy to see $[tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}}$ , whenever $s>r>0$ .