There is no useful formula known for this, unlike prime counts.

### 1: Introduction

We sieve for the answer, using some small table acceleration. Returns an approximation to the twin prime count of n. This returns quickly and has a very small error for large values. For inputs under 10M, a correction factor is additionally applied to reduce the mean squared error. Returns an array reference to semiprimes between the lower and upper limits inclusive , with a lower limit of 4 if none is given. The semiprimes are composite integers which are products of exactly two primes. This works just like the "primes" function. Similar to prime count, but returns the count of semiprimes composites with exactly two factors.

A fast method that requires computation only to the square root of the range end is used, unless the range is so small that walking it is faster. Returns an approximation to the semiprime count of n. This returns quickly and is typically square root accurate. Generating Ramanujan primes takes some effort, including overhead to cover a range. This will be substantially slower than generating standard primes. Similar to prime count, but returns the count of Ramanujan primes.

Given a start value n , and native unsigned integers width and depth , a sieve of maximum depth depth is done for the width consecutive numbers beginning with n. An array of offsets from the start is returned. Efficiently finds prime clusters between the first two arguments low and high. The remaining arguments describe the cluster. The cluster values must be even, less than 31 bits, and strictly increasing. Above this range, all cluster values are BPSW probable primes no counterexamples known. This function returns an array rather than an array reference.

Typically the number of returned values is much lower than for other primes functions, so this uses the more convenient array return. The cluster is described as offsets from 0, with the implicit prime at 0.

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The list 2,6,8 will find prime quadruplets. Note that there is no requirement that the list denote a constellation a cluster with minimal distance -- the list 42,92, is just fine. Returns the summation of primes between the lower and upper limits inclusive , with a lower limit of 2 if none is given. This is essentially similar to either of:. The current implementation is a small-table-enhanced sieve count for sums that fit in a UV, an efficient sieve count for small ranges, and a Legendre sum method for larger values.

While this is fairly efficient, the state of the art is Kim Walisch's primesum.

It is recommended for very large values, as it can be hundreds of times faster. With a single argument this prints all primes from 2 to n to standard out. With two arguments it prints primes between low and high to standard output. With three arguments it prints primes between low and high to the file descriptor given.

It will produce identical output to:. The point of this function is just efficiency. It is over 10x faster than using say , print , or printf , though much more limited in functionality. A later version may allow a file handle as the third argument. Returns the prime that lies in index n in the array of prime numbers.

Like most programs with similar functionality, this is one-based. For relatively small inputs below 1 million or so , this does a sieve over a range containing the nth prime, then counts up to the number. This is fairly efficient in time and memory. For larger values, create a low-biased estimate using the inverse logarithmic integral, use a fast prime count, then sieve in the small difference.

While this method is thousands of times faster than generating primes, and doesn't involve big tables of precomputed values, it still can take a fair amount of time for large inputs.

Think about whether a bound or approximation would be acceptable, as they can be computed analytically. If the result is larger than a native integer size bit or bit , the result will take a very long time. Returns an analytical upper or lower bound on the Nth prime. No sieving is done, so these are fast even for large inputs. For tiny values of n. For small inputs, an inverse of the opposite prime count bound is used.

For larger values, the Dusart and Axler bounds are used. For larger values, the inverse logarithmic integral is used.

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The value returned will not necessarily be prime. This applies to all the following nth prime approximations, where the returned value is close to the real value, but no effort is made to coerce the result to the nearest set element. Returns the Nth twin prime. This is done via sieving and counting, so is not very fast for large values. Returns an approximation to the Nth twin prime.

A curve fit is used for small inputs under , while for larger inputs a binary search is done on the approximate twin prime count. Returns the Nth semiprime, similar to where a forsemiprimes loop would end after N iterations, but much more efficiently. Returns an approximation to the Nth semiprime. Returns the Nth Ramanujan prime. For reasonable size values of n , e. If multiple calls are being made, it will be much more efficient to get the list once.

Takes a positive number n and one or more non-zero positive bases as input. Returns 1 if the input is a probable prime to each base, 0 if not.

## Sieve theory

This is the simple Fermat primality test. Note that there is a set of composites the Carmichael numbers that will pass this test for all bases. This downside is not shared by the Euler and strong probable prime tests also called the Solovay-Strassen and Miller-Rabin tests. Returns 1 if the input is an Euler probable prime to each base, 0 if not.

This is the Euler test, sometimes called the Euler-Jacobi test.

## Prime-detecting Sieves - Glyn Harman, J. H. Coates - Google Books

If 0 is returned, then the number really is a composite. If 1 is returned, then it is either a prime or an Euler pseudoprime to all the given bases. Given enough distinct bases, the chances become very high that the number is actually prime. This test forms the basis of the Solovay-Strassen test, which is a precursor to the Miller-Rabin test which uses the strong probable prime test. There are no analogies to the Carmichael numbers for this test. Returns 1 if the input is a strong probable prime to each base, 0 if not. If 1 is returned, then it is either a prime or a strong pseudoprime to all the given bases.

Given enough distinct bases, the chances become very, very high that the number is actually prime. This is usually used in combination with other tests to make either stronger tests e. Given the small chances of passing multiple bases, there are some math packages that just use multiple MR tests for primality testing.

Even inputs other than 2 will always return 0 composite.