PRP test for Mersenne numbers

The Great Internet Mersenne Prime Search is searching for new largest prime numbers.

A Mersenne candidate (which may be prime or not) is of the form M(E)=2^E-1 where the exponent E is a prime number itself. For example, the largest prime number known right now is 2^82'589'933-1 discovered in 2018.

The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. For Mersenne candidates a common choice for the base is 3. The Fermat test would then check whether: 3^p-1 =?= 1 where equality indicates that p is probable prime, while non-equality indicates that p is composite.

When p is a Mersenne number (i.e. p==2^E-1) a variant of the test becomes more convenient: 3^p+1 == 3^2E =?= 9

This (probable) primality test for Mersenne numbers, that I call PRP or PRP(3), is particularly convenient as it can be implemented, for exponent E, as a sequence of E modular squarings (called PRP iterations).

As an example, let's run the PRP test for M(7)==2⁷-1==127. Starting with A==3 we repeatedly apply the iteration A=A² modulo 127.

iteration	residue (modulo 127)
1	9
2	81
3	84
4	71
5	88
6	124
7	9

Because the final residue is 9 (the square of our starting value A==3) the outcome of this PRP test is probable prime.

Let's now run the test for M(11)==2¹¹-1==2047.

iteration	residue (modulo 2047)
1	9
2	81
3	420
4	358
5	1250
6	639
7	968
8	1545
9	223
10	601
11	929

Because the final residue is 929!=9 the outcope of this PRP test is composite.