Introduction to primesFirst, some basic commands. 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JSFHQzE+SSJORzYiIlMhKnljTTchKnljTTchKnljTTchKnljTTchKnljTTciIiI+SSpudW1wcmltZXNHRiUiIiEhIiI+SS5udW1jb21wb3NpdGVzR0YlRipGKz8oSSJpR0YlRidGJyImKysiSSV0cnVlRyUqcHJvdGVjdGVkR0AlLy1JKGlzcHJpbWVHNiRGMkkoX3N5c2xpYkdGJTYjLCZGJEYnRi9GJ0YxPkYpLCZGKUYnRidGJz5GLSwmRi1GJ0YnRidGK0YpRidGLUYnLUkmZXZhbGZHRjI2IywkRikjRidGMEYnLUZANiMqJC1JI2xuR0YlNiNGJEYrprint(); # input placeholderSo we see the actual proportion of primes is slightly higher but roughly the same as the 1/log(N) proportion predicted by the Prime Number Theorem. Another example with an 85 digit number:QyU+SSJORzYiLUkiXkclKnByb3RlY3RlZEc2JCIiKCIkKyIiIiItSSZldmFsZkdGKDYjLUkmbG9nMTBHNiRGKEkoX3N5c2xpYkdGJTYjRiQ=print(); # input placeholderQy8+SSpudW1wcmltZXNHNiIiIiEhIiI+SS5udW1jb21wb3NpdGVzR0YlRiZGJz8oSSJpR0YlIiIiRiwiJisrIkkldHJ1ZUclKnByb3RlY3RlZEdAJS8tSShpc3ByaW1lRzYkRi9JKF9zeXNsaWJHRiU2IywmSSJOR0YlRixGK0YsRi4+RiQsJkYkRixGLEYsPkYpLCZGKUYsRixGLEYnRiRGLEYpRiwtSSZldmFsZkdGLzYjLCRGJCNGLEYtRiwtRj42IyokLUkjbG5HRjQ2I0Y4Ric=print(); # input placeholderWe now want to explore the average number of prime factors for large numbers. The general theory claims the number of prime factors of N should be roughly log(log(N)). We will view some factorizations just to get an idea how many factors there are. Warning: Maple factors numbers up to 50 digits fairly well, but if you go past 60 or 70 digits, it is extremely slow. 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