The proof for the Ludqw Lehmer test was a little difficult for me to understand. I did not follow the proof very well. It was bouncing around in all sorts of areas, and in general just hard to see where it was going. However, it was rather interesting how it brought in so many different concepts.
Another part that was interesting was how it revolves around the Mersenne primes. They seem to be a particularly useful breed of primes. Also, it seems that they would be 3 mod 4 which we discussed in cryptography to be a desirable trait in a prime number.
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