[Request] How long to test this prime?

Double Mersenne primes are given by this series 2^(c) -1 = c₂, where the next term is then 2^(c)₂ -1 = c₃.

Starting with C₁ = 0 or 1 leads to an infinite loop of trivial 0s or 1s respectively. If instead we begin with c₁ = 2 this chain yields the results:

2^(2) - 1 = 3

2^(3) - 1 = 7

2^(7) - 1 = 127

2^(127) - 1 = 170,141,183,460,469,231,731,687,303,715,884,105,727

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These are all prime numbers.

The next number in the sequence would be:

2^(170,141,183,460,469,231,731,687,303,715,884,105,727) - 1

It is not yet known if this is prime given it's absolutely mindboggling enormity. Determining the primality of the 5^(th) term in this sequence would be the next step involved in proving the Catalan Conjecture.

Given the current speed of primality tests, how long would it take to test this number if we diverted all global computing power, from super computers to smart toasters, to checking if this is actually a prime?

More information about this sequence can be found on it's entry in the OEIS.