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`isprime`

Primality test

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Syntax

```isprime(`n`)
```

Description

`isprime(n)` checks whether `n` is a prime number.

`isprime` is a fast probabilistic prime number test (Miller-Rabin test). The function returns `TRUE` when the positive integer`n` is either a prime number or a strong pseudo-prime for 10 independently and randomly chosen bases. Otherwise, `isprime` returns `FALSE`.

If `n` is positive and `isprime` returns `FALSE`, then `n` is guaranteed to be composite. If `n` is positive and `isprime` returns `TRUE`, then `n` is prime with a very high probability.

Use `numlib::proveprime` for a prime number test that always returns the correct answer. Note, however, that it is usually much slower than `isprime`.

`isprime()` and `isprime(1)` return `FALSE`. `isprime` returns always `FALSE` if `n` is a negative integer.

`isprime` returns an error message if its argument is a number but not an integer. `isprime` returns a symbolic `isprime` call if the argument is not a number.

Examples

Example 1

The number `989999` is prime:

`isprime(989999)`

`ifactor(989999)`

In contrast to `ifactor`, `isprime` can handle large numbers:

`isprime(2^(2^11) + 1)`

`isprime()` and `isprime(1)` return `FALSE`:

`isprime(0), isprime(1)`

Negative numbers yield `FALSE` as well:

`isprime(-13)`

For non-numeric arguments, a symbolic `isprime` call is returned:

`delete n: isprime(n)`

Parameters

 `n` An arithmetical expression representing an integer

Return Values

Either `TRUE` or `FALSE`, or a symbolic `isprime` call.

Algorithms

Reference: Michael O. Rabin, Probabilistic algorithms, in J. F. Traub, ed., Algorithms and Complexity, Academic Press, New York, 1976, pp. 21–39.