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 Post subject: Question?
PostPosted: Thu, 19 Aug 2010 09:22:22 UTC 
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Let A1,A2.....An be n numbers such that A2 is either 1 or -1.if A1A2A3A4+A2A3A4A5......+AnA1A2A3=0 then prove that 4 divides n. (1,2... are in base of A)


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 Post subject: Re: Question?
PostPosted: Thu, 19 Aug 2010 10:17:44 UTC 
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anmol singh wrote:
Let A1,A2.....An be n numbers such that A2 is either 1 or -1.if A1A2A3A4+A2A3A4A5......+AnA1A2A3=0 then prove that 4 divides n. (1,2... are in base of A)


Thought I have seen this question before, possibly on another board. You mean A_i=\pm 1 for all i=1,2,\dots,n, not just A_2.

Spoiler:
Easy to get 2|n. To get 4|n, consider the effect of changing signs on the sum.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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