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binomial coefficient

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outermeasure

Post subject: binomial coefficient

Posted: Mon, 11 Dec 2017 15:20:43 UTC

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(1) Let p be a prime. What is $\displaystyle\binom{2p}{p}$ modulo p^2?

(2) Let p be an odd prime. What is $\displaystyle\binom{3p}{p}$ modulo p^3?

Hint:

Spoiler:

Big hint which trivializes the problem:

Spoiler:

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

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