PARI's commands - Page 139

science_man_88 · 2010-10-24, 20:31

no Pi I'm working on about Mersenne numbers.

CRGreathouse · 2010-10-24, 21:32

Quote:

Originally Posted by 3.14159

I have a challenge for you all;

k * n! + 1 = p², where p is a prime integer, and where k ≤ n!

Ex: 5 * 8! + 1 = 449²

I find none with 8 < n < 10,000. I don't expect that any exist with larger n; the heuristic probability of finding one is less than 10^-17825.

3.14159 · 2010-10-24, 21:47

Quote:

Originally Posted by CRGreathouse

I find none with 8 < n < 10,000. I don't expect that any exist with larger n; the heuristic probability of finding one is less than 10^-17825.

How did you arrive at those figures?

CRGreathouse · 2010-10-24, 21:57

Quote:

Originally Posted by 3.14159

How did you arrive at those figures?

For n <= 1e4: direct search; for n > 1e4: numerical integration.

The same technique suggests about 4.9 below 1e4, which is a pretty good estimate (there are 6).

3.14159 · 2010-10-24, 23:12

What are the odds of a 210-digit odd number being a p105 * p105?

I'm guessing from 0.001% to 0.1%.

CRGreathouse · 2010-10-24, 23:34

Quote:

Originally Posted by 3.14159

What are the odds of a 210-digit odd number being a p105 * p105?

An easy upper bound is 0.00155%. A closer analysis gets 0.00127419%, though my calculations on the latter assume the Riemann hypothesis.

3.14159 · 2010-10-24, 23:53

Quote:

Originally Posted by CRGreathouse

An easy upper bound is 0.00155%. A closer analysis gets 0.00127419%, though my calculations on the latter assume the Riemann hypothesis.

There's those unproven conjectures again..

CRGreathouse · 2010-10-25, 00:02

Quote:

Originally Posted by 3.14159

There's those unproven conjectures again..

Thus my first answer, which does not assume it. I could have given a better unconditional answer, but that would be too much work.

cmd · 2010-10-25, 00:23

ep

the right mathematics can not cheat

3.14159 · 2010-10-25, 00:24

Quote:

Originally Posted by CRGreathouse

Thus my first answer, which does not assume it. I could have given a better unconditional answer, but that would be too much work.

Hopefully it would be accurate..

3.14159 · 2010-10-25, 00:25

Quote:

Originally Posted by cmd

ep

Hey, can you try spamming another thread, bozo?

2010-10-24, 20:31	#1519
science_man_88 "Forget I exist" Jul 2009 Dumbassville 2⁶·131 Posts	no Pi I'm working on about Mersenne numbers. Last fiddled with by science_man_88 on 2010-10-24 at 20:31

2010-10-24, 23:12	#1523
3.14159 May 2010 Prime hunting commission. 11010010000₂ Posts	What are the odds of a 210-digit odd number being a p105 * p105? I'm guessing from 0.001% to 0.1%. Last fiddled with by 3.14159 on 2010-10-24 at 23:15

2010-10-25, 00:23	#1527
cmd "(^r'°:.:)^n;e'e" Nov 2008 ;t:.:;^ 3³×37 Posts	ep the right mathematics can not cheat Last fiddled with by cmd on 2010-10-25 at 00:28 Reason: i t e p

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