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Originally Posted by BrianE
(1) Discover new Mersenne Primes
(2) Prove that there are no lower Mersenne Primes than the ones already discovered

Or, you could combine (1) & (2) like: "Discover new Mersenne Primes in exhaustive search through all untested candidates."
Some earlier Mersennehunting efforts tested only certain ranges and skipped others, so were nonexhaustive.
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(3) For composite Mersenne numbers with accessible factor(s) find the lowest factor

That is not actually a GIMPS goal.
In GIMPS, factoring has been only secondary, performed only because it was the most efficient way, up to a certain point, to eliminate candidates without performing the costly LucasLehmer computation.
Early in the project, finding the lowest factor was a convenient sideeffect of this primecandidateelimination effort. It does have some mathematical value. So, the trialfactoring portion of the software, which for reasons of efficiency searches for factors somewhat outoforder (though eventually checking all possibilities), would, upon finding a factor, spend a small amount of extra time determining whether there was also a slightly smaller factor or factors that would have been found if candidates had been tried in strictly sequential order. That guaranteed that the smallest factor was found.
Unfortunately, there was a software bug that sometimes caused that extra endsearch to be skipped. Once it was discovered, it meant that some earlier GIMPS factoring results were
not guaranteed to be the lowest factors. It was decided to omit that extra endsearch for all efforts beyond a certain point. So, currently, factors found are not guaranteed to be the lowest.
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(1) and (3) are clearly relatively easy to verify by the Mathematical community because they involve individual Mersenne numbers. But what about (2)?

Well, each individual candidate is easily verifiable there, too. It is true that there are a much larger number of candidates to be verified for (2) than for (1), but that's just a matter of allocating sufficient resources, not a matter of actual impossibility.
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It is not really independently verifiable that what GIMPS now calls M39 is the 39th Mersenne because noone can run tens of thousands of computers for another 10 years.

Yes, it
is possible to do independent verification! What GIMPS did in its first 10 years takes less and less time to independently verify as computers get faster. The CPU I now use is over 40 times as fast as the one I used my first five years in GIMPS. I (or anyone else) could reperform my first five years of GIMPS work in a couple of months now, using independentlywritten software if so wished to serve as verification.
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Are the safeguards (the double checking of Lucas Lehmer tests and requirement of two tests with the same final residue) generally accepted as proof by the general Mathematical community

Welcome to the era of experimental computational mathematics, in which not all results are absolutely proven certain beyond any shadow of a doubt. (Actually, that has always been true. Ask Pythagoras about the square root of 1. Ask someone who's found a flaw in a proof that had been generally accepted for decades.) We can only do our best to check for errors, including error analysis similar to that used in other experimental science. This has been discussed many times in this forum.
Another example of what I'm calling experimental computational mathematics was the 1977 proof of the FourColor Theorem. It was done by computer program, it had the possibility of hardware or software error, and so was not _absolutely_ proven. (But as of 2005 another, independent proof seems to have been more certainly verified.) Indeed, the FourColor Theorem had a precomputer history of published "proofs" that turned out to have flaws! See the MathWorld entry at
http://mathworld.wolfram.com/FourColorTheorem.html.