Find the order of $U_N$ for $N=p_1*p_2*p_3$ such that $p_i$ is prime. $U_N$ are all the integers relatively prime to $N$.
I believe that this is simply adding up each $p_i$ in this case. It works for 2,3, and 5, but if my memory serves me right this is a special group that I forget about. The next smallest set of 3 primes is 2,3, and 7. But that is quite a lot to do by brute force. Can somebody point me in the right direction if they have seen this anywhere else?
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asked Apr 12, 2015 at 21:35
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1$\begingroup$ What is your definition of $U_N$? $\endgroup$– QuditApr 12, 2015 at 21:42
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$\begingroup$ edited. my apologizes $\endgroup$– Jack ArmstrongApr 12, 2015 at 21:44
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1 Answer
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Note that $|U(n)|=\phi(n)$ where $\phi$ is Euler's totient function.
For $n=p_1p_2p_3$ this gives $$\phi(n)=\phi(p_1p_2p_3)=\phi(p_1)\phi(p_2)\phi(p_3)=(p_1-1)(p_2-1)(p_3-1)$$
Assuming $p_i\neq p_j$ for $j\neq i$.
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$\begingroup$ right. I was doing $U_{30}$ as 10 not 8. thanks $\endgroup$ Apr 12, 2015 at 21:50