PARI's commands

[CRGreathouse]

Originally Posted by 3.14159

So: Anyone else up for the challenge of posting a cofactor prime of at least 500 digits? (Note: The cofactor has to be of one of the 19 types of prime listed.)

By your standards and with my usual method, I find a p501 in 203*2^1658+1.

[3.14159]

Originally Posted by CRGreathouse

By your standards and with my usual method, I find a p501 in 203*2^1658+1.

Let me guess: 3 * p501.

Okay, wiseguy, the smallest factor must be a p40, and the digits must follow a random distribution that is statistically likely. Good luck.

[3.14159]

Let's see how you loophole your way out of that one!

[CRGreathouse]

Originally Posted by 3.14159

Let me guess: 3 * p501.

Okay, wiseguy, the smallest factor must be a p40, and the digits must follow a random distribution that is statistically likely. Good luck.

No thanks, my cores are busy extending several sequences relating to pseudoprimes and large prime gaps. I do enjoy finding trivial examples, though.

[CRGreathouse]

Originally Posted by 3.14159

Let's see how you loophole your way out of that one!

One possibility: go through 20-digit primes, finding k and n values that give a (521 to 540)-digit multiple of that prime, checking if the cofactor is a 2-pseudoprime. This should be faster than actually factoring them out. But I'm not going to do this search because it's just as silly as it sounds.

[3.14159]

Originally Posted by CRGreathouse

No thanks, my cores are busy extending several sequences relating to pseudoprimes and large prime gaps. I do enjoy finding trivial examples, though.

Ah, but you could have loopholed your way out of this one, as I made no digit requirement. It's as easy as:

1230021358933920151693112553959118376943838 * 400³⁵ + 1 = 8826736195555687888973540222790863753879 * 1645175382483954577575076564167802963870136473616621363381331706098141318612978687213529959719.

And, voila! You would have won, yet again.

[CRGreathouse]

Originally Posted by 3.14159

Never said the divisors had to be small

I suppose you could have cofactor (small factor) and cofactor (large factor) categories. It would be interesting to properly classify the difficulty of finding (1) a general and (2) a special-form cofactor based on the size of both the small and large factor. The main difficulty is the latter, of course, but the former plays a part too.

[3.14159]

Originally Posted by CRGreathouse

One possibility: go through 20-digit primes, finding k and n values that give a (521 to 540)-digit multiple of that prime, checking if the cofactor is a 2-pseudoprime. This should be faster than actually factoring them out. But I'm not going to do this search because it's just as silly as it sounds.

See above

[CRGreathouse]

Originally Posted by 3.14159

See above

I beat you to it.

[3.14159]

Originally Posted by CRGreathouse

I suppose you could have cofactor (small factor) and cofactor (large factor) categories. It would be interesting to properly classify the difficulty of finding (1) a general and (2) a special-form cofactor based on the size of both the small and large factor. The main difficulty is the latter, of course, but the former plays a part too.

A special-form cofactor in a special-form number? Please, this only happens in Mersenne numbers. If you wanted a special-form factor, you would have to directly rig it to do so.

[CRGreathouse]

Originally Posted by 3.14159

the digits must follow a random distribution that is statistically likely

How would you define this, I wonder?