New prime record: 51st Mersenne prime

I think there's a simple calculation to estimate that the expected financial value of running Prime95 (just for this particular prize) is negative.

Suppose GIMPS as a whole discovers one new Mersenne prime per year and pays its discoverer $3000. According to https://www.electricchoice.com/electricity-prices-by-state/, the average U.S. residental customer pays 13.11 cents per kilowatt-hour for electricity.

Suppose that you wanted to run a desktop computer with Prime95, hypothetically consuming 80 watts (which is probably low).

  You have: 13.11 cent/kWh
  You want: dollar/(80 watt*year)
  	* 91.935844

So it's going to cost about $91 to run that computer for a year, which means that your net earnings are negative in expectation if your computational contribution to GIMPS from that machine is less than than 3000/91 or about 1/33 of the network's total computational power.

There are lots of ways to make this estimate better, for example by finding the actual fraction of computational capacity that an individual machine contributes, or the actual power consumption of that machine, or the actual cost of electricity for that machine's owner, or a better estimate than "one per year" of the GIMPS discovery rate (for example, their current press release says that recently they've found about three times as many primes as they expected).

Interestingly there are only 6 known primes of the form 2^n+1. Other than "2" these are all Fermat primes of the form 2^(2^n)+1 and so there is only a single known case n is prime - the case of n=2.

https://oeis.org/A092506 https://oeis.org/A019434

Oh my.