Algebra

Elementary symmetric of a sum of matrices ★★★

Author(s):

Problem  

Given a Matrix $ A $, the $ k $-th elementary symmetric function of $ A $, namely $ S_k(A) $, is defined as the sum of all $ k $-by-$ k $ principal minors.

Find a closed expression for the $ k $-th elementary symmetric function of a sum of N $ n $-by-$ n $ matrices, with $ 0\le N\le k\le n $ by using partitions.

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Posted by rscosa
updated August 27th, 2010
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Number Theory » Combinatorial N.T.

Frobenius number of four or more integers ★★

Author(s):

Problem   Find an explicit formula for Frobenius number $ g(a_1, a_2, \dots, a_n) $ of co-prime positive integers $ a_1, a_2, \dots, a_n $ for $ n\geq 4 $.

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Posted by maxal
updated November 26th, 2008
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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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Number Theory

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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Group Theory

Inverse Galois Problem ★★★★

Author(s): Hilbert

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

Keywords:

Posted by tchow
updated October 13th, 2008
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