Primes
This section contains functions for generating primes and analyzing primality.
Prime number generation
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Returns all primes \(p\) for \(p \le n\). |
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Returns the \(k\)-th prime. |
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Returns the nearest prime \(p\), such that \(p \le n\). |
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Returns the nearest prime \(p\), such that \(p > n\). |
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Returns a random prime \(p\) with \(b\) bits, such that \(2^b \le p < 2^{b+1}\). |
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Returns all known Mersenne exponents \(e\) for \(e \le n\). |
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Returns all known Mersenne primes \(p\) for \(p \le 2^n - 1\). |
Primality tests
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Determines if \(n\) is prime. |
Determines if \(n\) is a prime power \(n = p^k\) for prime \(p\) and \(k \ge 1\). |
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Determines if \(n\) is a perfect power \(n = x^k\) for \(x > 0\) and \(k \ge 2\). |
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Determines if \(n\) is composite. |
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Determines if the positive integer or the non-constant, monic polynomial is square-free. |
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Determines if the positive integer \(n\) is \(B\)-smooth. |
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Determines if the positive integer \(n\) is \(B\)-powersmooth. |
Specific primality tests
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Determines if \(n\) is composite using Fermat's primality test. |
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Determines if \(n\) is composite using the Miller-Rabin primality test. |