- **Factoring**
(*https://www.mersenneforum.org/forumdisplay.php?f=19*)

- - **Exponent fully factored whilst only 74% known**
(*https://www.mersenneforum.org/showthread.php?t=21503*)

Exponent fully factored whilst only 74% knownCan someone explain why [URL="http://www.mersenne.ca/exponent/397"]M397[/URL] is labeled as fully factored, when the details are:
[QUOTE]Known prime factors (8 factors, 294.6 bits, 74.21164755% known): Remaining cofactor is a probable-prime[/QUOTE] Why wouldn't the exponent need to be 100% known before it can be labeled fully factored? I also note that there are no PRPs attached to [URL="http://www.mersenne.ca/prp.php?show=2&min_exponent=0&max_exponent=1000#M397"]M397[/URL], which I would have expected with the mention of a probably-prime. |

[QUOTE=mattmill30;439983]Can someone explain why [URL="http://www.mersenne.ca/exponent/397"]M397[/URL] is labeled as fully factored[/QUOTE]
Two answers. 1) Because it is. 2) Lazy programmer. In this particular case, all the factors are small and proven prime, but in the general case, a fully-factored exponent will result from a bunch of small factors and a large PRP cofactor. We're confident that the PRP cofactor is prime, but it will not be mathematically proven. There is no separate logic in the website to handle when the last factor is proven vs PRP. |

[QUOTE=mattmill30;439983]Can someone explain why [URL="http://www.mersenne.ca/exponent/397"]M397[/URL] is labeled as fully factored, when the details are:
Why wouldn't the exponent need to be 100% known before it can be labeled fully factored? I also note that there are no PRPs attached to [URL="http://www.mersenne.ca/prp.php?show=2&min_exponent=0&max_exponent=1000#M397"]M397[/URL], which I would have expected with the mention of a probably-prime.[/QUOTE]You can verify the final factor 6597485910270326519900042655193 quite easily. [size=1]It is a real prime BTW[/size] |

By convention, if an exponent is fully factored, the largest factor is omitted from the database (in both mersenne.org and mersenne.ca).
Storing it would be redundant, since it easily calculated by dividing the Mersenne number itself by all of its other factors. And in most cases (although not here), it is many orders of magnitude larger than the next largest factor, so there would be enormous storage costs, for example: M5240707 = 75392810903 * (a probable prime with 1.5776 million digits) Note that the 273 fully factored Mersenne exponents up to and including M63703 really are fully factored, with primality certificates calculated for the remaining cofactor using programs like Primo, whereas all larger "fully factored" Mersenne exponents (of which 30 are known) are only "probably fully factored". |

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