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A^2+B^2=C^2 and A^4+B^4=C^4 - OIMU 2008 Problem 6

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August 28, 2010
linear algebramatrixmodular arithmeticlinear algebra unsolved

Problem Statement

a) Determine if there are matrices A,B,C∈SL2(Z)A,B,C\in\mathrm{SL}_{2}(\mathbb{Z}) such that A2+B2=C2A^2+B^2=C^2.
b) Determine if there are matrices A,B,C∈SL2(Z)A,B,C\in\mathrm{SL}_{2}(\mathbb{Z}) such that A4+B4=C4A^4+B^4=C^4.
Note: The notation A∈SL2(Z)A\in \mathrm{SL}_{2}(\mathbb{Z}) means that AA is a 2×22\times 2 matrix with integer entries and det⁡A=1\det A=1.
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