Or when you can't afford a retrofit of your system with a heat pump.

Or when you need to target dry a small room or a pair of wet shoes.

They have plenty of other uses, the only caveat I guess is that they aren't used as the main source of heat.

That's why I proposed encasing them in a heat transfer shell and keep the thing submerged in water. They could keep the water in the boiler at around 60 Celsius.

At what temperature would a heat pump be less efficient than an electric resistance?

Soil, at a depth of only around a few meters, rarely exceeds 10C, even in summer. That gives you a 45C differential for your boiler in the worst of worst cases with a ground heat exchanger, which is more than enough for best-case efficiency, year long.

This will be less expensive than a crypto in material cost very rapidly.

That I never doubted.

There is no such temperature, with a good enough design of heat pump. A good heat pump is always more efficient at heating than electrical resistance.

The improvment is a multiplication factor, which can be quite large; it's not just a few percent. The factor is why I'd describe mining - or resistive heating - as a very inefficient use of electricity for heating.

Since we are theorising about the merits of miners, we may as well use the theoretical limit for the alternative. Wikipedia explains: "If the low-temperature reservoir is at a temperature of 270 K (−3 °C) and the interior of the building is at 280 K (7 °C) [...] only 1 joule of work [...] 28 joules of heat are added to the building interior".

In other words, a heat pump can be 28 times more efficient than running a miner or other resistive electrical heater in those conditions. But that's a high number, a theoretical limit in conditions chosen to put heat pumps in their best light.

More realistic, based on real devices: "In Europe, the standard test conditions for ground source heat pump units use 35 °C (95 °F) [...] and 0 °C (32 °F) [...] the maximum theoretical COPs would be [...] 8.8. Test results of the best systems are around 4.5. When measuring installed units over a whole season and accounting for the energy needed to pump water through the piping systems, seasonal COP's for heating are around 3.5 or less. This indicates room for further improvement."

So 3.5-4.5 times more efficient than running a miner for heating in real conditions with current heat pumps, which can be improved. That translates directly to reduced atmospheric emissions from electricity generation.

The efficiency of the heat pump reduces when the outside temperature reduces, and reduces when the inside temperature increases. Conversely, if you are heating water, the time when the heat pump is most efficient is in the summer.

Cost is distinct from efficiency. The cost of a unit of energy as electricity is higher than the same unit of energy as gas, for example, so if you are comparing an electrical heat pump to gas heating, at a low enough outdoor temperature the gas heating is cheaper. This is why some places switch between heat pump and gas modes. The same argument does not apply when comparing with electrical heating like miners, or with an electrical water boiler.

That's a key point. If your power comes from renewables, this discussion is a lot less interesting. I'm all in for efficient use of energy (and I don't think crypto mining is a good use of it), but I also know a heat pump is a lot more complicated than an electric resistance (or a hot PCB board).