For the purposes of this question, we will assume that they both landed successfully. Ingenuity was deployed properly. Then, Perseverance drove away, and then Perserverance say hit a large rock and failed. Could Ingenuity survive on its own? Or does it rely on Perserverance?
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No. Perseverance is the communications relay for Ingenuity.
Ingenuity uses essentially ZigBee to communicate, the same protocol that possibly your smart bulbs use. ZigBee has a range of about 300m line-of-sight without obstructions which is just a tiny bit too short for communicating with Earth.
Jörg W Mittag
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16Ingenuity pushes a bit more power than a typical ZigBee and "The link is designed to relay data at over-the-air rates of 20 kbps or 250 kbps over distances of up to 1000 m**" which is still a bit too short of Earth – BrendanLuke15 May 23 '22 at 21:16
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16Indeed. Note that my answer is a bit tongue-in-cheek, since of course Perseverance also (usually) does not communicate with Earth (although it can in a pinch) but with one of the Martian relays. They're not quite as far away as Earth, but still orders of magnitude farther than 1000m. – Jörg W Mittag May 23 '22 at 21:24
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3Imagine a whole planet free of EM noise-generating devices! – fraxinus May 24 '22 at 09:01
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4All the unfortunate sub-surface Martian homes finding their smart bulbs going haywire :( – dbmag9 May 24 '22 at 11:39
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1As we say on explainxkcd.com: {{citation needed}} – Barmar May 24 '22 at 14:24