Why doesn't this congruence have solutions $x^2=83 \pmod{83^{2000}}$
I know $x^2=83 \pmod{83}$ has a solution, I'm not sure how to show the above congruence doesn't though.
2026-03-29 03:28:32.1774754912
Why doesn't this congruence have solutions $x^2=83 \pmod{83^{2000}}$
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Hint $\ \color{green}{x^2} = p + p^2n = \color{#c00}p(1+pn)$ has $\color{green}{\,\rm evenly}\,$ many $p$'s on the left, but $\color{#c00}{\rm\,oddly}\,$ many on the right (since $\,p\nmid 1+pn),\,$ when their (unique!) prime factorizations are considered, for $\,p\,$ prime.