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I was under the impression that if you hold your Bitcoin you don't pay taxes on gains, unless you cash out. Is this accurate?

Source: https://bitcoin.tax/faq

Do I need to file or pay taxes if I own Bitcoins?

Not if you just bought Bitcoins (or any crypto-currencies) with your own money. However, if you traded, sold, or used any to purchase something, then you might. If you were given Bitcoins as payment, as a salary, or as a gift/donation, this is income and should be reported as any other income you earn.

I didn't move any money out of the exchange, I just bought back in. What about then?

Yes. A tax event occurred and you gained money, even though it isn't in your bank account. For tax purposes it is treated no differently.

right. When you cash out you have to declare your gains. It is the case in France so I guess it is the same in many other countries.