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I'd be interested to know if the bitcoin network has ever been down. By "down" I mean that a client would not be able to synchronize with the network to obtain data like the txs in the mempool or recently mined blocks.
My guess would be that this would be extremely rare due to the decentralized architecture, but it'd be great to know for sure. If there have been downtimes, I'd be interested to know how frequent these have been and how long they have persisted.
A bitcoin node could be down or (extremely rarely as you say) all its peers could be down... but the bitcoin network could not be considered down; you could still synchronize by connecting to different peers manually using fallback nodes or even have your own list of nodes.
The only way for the bitcoin network to be down is if all the nodes+miners are down which would be nearly impossible even with a few dozens of nodes (current estimate is 6000+ nodes). However, in the very beginning, when there where 1 or 2-3 machines it was possible (see example in @Murch's answer, where that early on, there was only one node on the network: Satoshi's).
Yes. In the very beginning of the Bitcoin project there were long gaps between blocks that suggest that no node was mining/online at all. E.g. between between Block 27 and 28 there was a 8h34 pause.
Block 27: 2009-01-10 06:56:13
Block 28: 2009-01-10 15:30:57
I would guess that in the later years, there was never a time when you wouldn't have been able to get blocks from another node, except if you had been disconnected from the Internet yourself.
The only likely possibility that I see today for nodes to be online but unable to synchronize with the blockchain would be if the internet were to be completely disconnected between continents or nations for a time, also known as netsplits.
Silly thought: Even then, synchronization and confirmations could be achieved with largely increased latency when the blockchain data/transaction submissions would be transported via physical media e.g. pigeons with a USB stick. You shouldn't trust zero-confirmation transactions while being split-off, though. ;)
The closest thing I can think of is the March 2013 fork. While the network wasn't down per se, you couldn't use it reliably until the issue was resolved.
