Integers Relatively Prime to the First n Primes

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The probability that a prime number does not divide a natural number is . Hence, the probability that a natural number is relatively prime to all primes less than equals . Legendre proved that the large limit of this product is zero, meaning that the probability that a large random integer is a prime approaches zero. This Demonstration computes the probability that an integer is coprime with each of the first primes.

Contributed by: Oleksandr Pavlyk (March 2011)
Open content licensed under CC BY-NC-SA


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Riemann's zeta function is defined by . Legendre's theorem states that .



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