# examples of probable primes

To give an example of a probable prime^{} relative to a base: ${4}^{341233}-{3}^{341233}$ has passed preliminary primality tests relative to bases 2, 3, 5, 7, 11, 13 and 101. Its square root is approximately $2.3362\cdot {10}^{102721}$, which makes a conclusive primality test by trial division^{} in a reasonable time period impractical.

To give an example of a probable prime by a pattern: this pattern

$${2}^{2}-1=3,{2}^{3}-1=7,{2}^{7}-1=127$$ |

$${2}^{127}-1=170141183460469231731687303715884105727$$ |

suggests that ${2}^{170141183460469231731687303715884105727}-1$ might be a Mersenne prime^{}. But since this is larger than the largest known Mersenne prime ${2}^{30402457}-1$ (as of 2005), a Lucas-Lehmer test might take longer than the average human lifetime.

On the other hand, $123456789\cdot {10}^{123456789}+123456789$ is not a probable prime, because even though it is much larger than either of the probable primes given above, it is clearly divisible by ${3}^{2}$.

Title | examples of probable primes |
---|---|

Canonical name | ExamplesOfProbablePrimes |

Date of creation | 2013-03-22 15:53:49 |

Last modified on | 2013-03-22 15:53:49 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Example |

Classification | msc 11A41 |