You may have also noticed that the perfect numbers listed above (6, 28, 496, 8128) all end with either the digit 6 or the digit 8--this is also very easy to prove (but no, they do not continue to alternate 6, 8, 6, 8,...). If you like that digit pattern, look at the first four perfect numbers in binary: 110 11100 111110000 1111111000000 (The binary digit pattern is a consequence of Theorem One.) It is not known whether or not there is an odd perfect number, but if there is one it is big! This is probably the oldest unsolved problem in all of mathematics.
Because a prime number has only the trivial factors 1 and , in his The Road Ahead , Bill Gates accidentally referred to a trivial operation when he stated "Because both the system's privacy and the security of digital money depend on encryption, a breakthrough in mathematics or computer science that defeats the cryptographic system could be a disaster. The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers [emphasis added]" (Gates 1995, p. 265).
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