4

Is there a list, similar to prime numbers and probable primes, of the largest semiprimes with unknown factors? Is there a list of numbers that are either semiprime or prime, with no known factors? Is there literature on how to find large semi primes?

Mathphile
  • 2,447
  • 15
  • 35
Tejas Rao
  • 2,075
  • I feel like a list of "possible semiprimes" would likely be at least somewhat coincident with a list of possible primes, given that a large semiprime accordingly has large prime factors (so finding said factors would be difficult, thus rendering the number a possible candidate for either). As for finding large semiprimes, I feel the easiest method would just be to multiply two large known primes together, but I imagine you want something more elegant. – PrincessEev Dec 08 '18 at 22:27
  • 1
    Maybe http://physics.open.ac.uk/~dbroadhu/cert/semgpch.gp is of interest. – gammatester Dec 08 '18 at 22:34
  • 1
    It's unclear how one could be certain that a large number was a semiprime (as opposed to having three or more prime factors) without knowing the two prime factors which multiply to it. – Keith Backman Dec 09 '18 at 02:33
  • 1
    @KeithBackman Surprisingly, this seems to be possible. I heard from a construction of a number which could be proven to be semiprime without known factors (according to the claim, not even by the constructor himself). For huge number, this method is probably not feasible. – Peter Dec 09 '18 at 14:11
  • 1
    What is by the way meant with "known factors" ? If someone creates a monster semiprime with a computer program by simply multiplying two huge primes the program does not display , then noone knows the factors of the resulting number. Does that count , or is it "cheating" ? – Peter Sep 21 '22 at 15:28
  • 2
    @Peter The idea is ‘what is the largest number that can be proven semiprime without knowing its factors.’ So in theory someone could multiply two very large primes, but proving that it a semiprime without reference to said factors is very non-trivial. – Tejas Rao Oct 03 '22 at 20:40
  • 1
    The link regarding a 5061-digit proven semiprime (due to David Broadhurst) posted by @gammatester seems broken (at least for me, I get "Forbidden"). Fortunately there was an earlier Question which covered the topic of constructing certified semiprimes without revealing their prime factors. See also this Question on how to check if a number is semiprime and Ed Pegg's Answer there. – hardmath Oct 11 '22 at 16:00

2 Answers2

2

As mentioned, this is apparently possible using some trick with elliptical something or other.

That said, it's also easy to build a poor man's version. Take some $n$; test it for primality; if it's not prime, test all prime factors up to $n^{1/3}$ to see if any divide $n$; if none do, then this is a semiprime with two large but totally unknown factors; otherwise, increment $n$ and repeat until you find one.

Trevor
  • 6,144
  • 16
  • 39
  • 1
    The catch of the trial division is that a number we can check this way in a reasonable time is so small that factoring it would take only a little time. But you are right that as long noone does it the factors are "unknown". – Peter Sep 21 '22 at 15:26
-3

It doesn't exist. Any area of rectangle which sides are prime is a semiprime.

LAAE
  • 413
  • 2
    There are also infinite many primes, but nevertheless it makes sense to speak of the largest known prime. – Peter Dec 09 '18 at 14:13