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~~~
PaulHoule
>>There is no reason to make any damn decisions at all.

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Why not do all this in your favorite dynamic language?

Q:

Show that every finite positive integer has a square root

Show that every finite positive integer has a square root.
So far I understand I need to write $n$ as a product of prime numbers $p_{1},p_{2},p_{3},\ldots,p_{k}$ which gives $n=p_{1}^{a_{1}}p_{2}^{a_{2}}p_{3}^{a_{3}}\ldots p_{k}^{a_{k}}$. If I then take $2^{a_{1}}p_{2}^{a_{2}}p_{3}^{a_{3}}\ldots p_{k}^{a_{k}}$ I will have the square root of $n$.
What I don’t get is the condition $k\gt0$ and $\sum a_{i}\gt0$. Do I just have to look through all positive integers to see which ones have this property or is there a rule to follow?

A:

You need to reason about prime factorizations. If you want to prove that every prime $p$ has a square root in $\mathbb{Z}_p$ (the residue field at $p$), you can reason by induction on $p$, or you can just consider a prime factorization of $p$ as a product of primes congruent
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