Extrapolation: How long until the minimum megaprime?

[CRGreathouse]

Originally Posted by petrw1

We can narrow it down to somewhere between:

http://primes.utm.edu/primes/lists/all.txt

rank  description                    digits who  year comment
35  59*2^3408416-1                  1026038 L426 2010 
36  3139*2^3321905-1                 999997 L185 2008

Dusart shows that for x at least 396738, (x, x + x/(25 log^2 x)) contains at least one prime. So more precisely it's between 10^999999 and 1.00000000000000755 times that.

There's still no effective version of Baker-Harman-Pintz AFAIK (though they say it's possible, I guess it's just too much work for someone to have done it?) which would give a much tighter bound.

[science_man_88]

Originally Posted by Brian-E

Science Man, is there anything you notice about the numbers , , , ... ,, ... and can you prove the property for all n?

yes they are all 5 mod 6 that's how I got that it was 5 mod 6. 10 = 4 mod 6 4^2 = 4 mod 6 so 100 = 4^2 mod 6 = 4 , 1000 = 4^2 mod 6 = 4.

[science_man_88]

Originally Posted by Brian-E

Science Man, is there anything you notice about the numbers , , , ... ,, ... and can you prove the property for all n?

since you like giving things through PM I get you want me to look at 11,1001,100001, etc with even numbers of 0's and realize that they are multiples of 11.

[science_man_88]

Originally Posted by xilman

It's easy to show that 10^1000000 has 1000001 digits and so is much larger than the first megaprime.

I suggest that you try again.

Paul

P.S. 10^999999+1 is composite. Prove this.

so this is of the form 11x + 1 as 11 is odd 50% of these numbers end up being even. that's as far as I know. but as it stands that all powers of 10 are 4 mod 6 we can conclude the n still stand for what to test. never mind no they don't I found an error.

[science_man_88]

n must actually be n

my error was jumping to 4,2 instead of 2,4 .

[petrw1]

From this same list: http://primes.utm.edu/primes/lists/all.txt

I can say for sure that we know the smallest prime with
100,000 digits?
10,000 digits?
1,000 digits?

[science_man_88]

forstep(x=1,100,[2,4],if(x%11!=1 && x%120!=79 && x%6!=2 && x%4!=3,print1(x",")))

so far i think I've figured this out, this left me with just 15 candidates below n=100.

[10metreh]

Originally Posted by petrw1

From this same list: http://primes.utm.edu/primes/lists/all.txt

I can say for sure that we know the smallest prime with
100,000 digits?
10,000 digits?
1,000 digits?

The smallest with 1000 digits is 10^999+7 - found in under a second and easily provable.
10^9999+33603 is on the list - I suspect that is the smallest with 10000 digits as I see no other reason for it to be on the list.

[science_man_88]

(17:49)>for(y=2,300,for(x=1,300,if(((10^x)%y)==((10^(x+1))%y),print(x" "y" "10^x%y);break())))
1 2 0
1 3 1
2 4 0
1 5 0
1 6 4
3 8 0
1 9 1
1 10 0
2 12 4
1 15 10
4 16 0
1 18 10
2 20 0
3 24 16
2 25 0
1 30 10
5 32 0
2 36 28
3 40 0
1 45 10
4 48 16
2 50 0
2 60 40
6 64 0
3 72 64
2 75 25
4 80 0
1 90 10
5 96 64
2 100 0
3 120 40
3 125 0
7 128 0
4 144 64
2 150 100
5 160 0
2 180 100
6 192 64
3 200 0
2 225 100
4 240 160
3 250 0
8 256 0
5 288 64
2 300 100

this can tell us loads up to x=300 ( though it doesn't have the range of y necessary) for example all 10^x for x>=2 are 40 mod 60 ( which is almost the same but stronger than 40 mod 120 for x>=3).

[CRGreathouse]

Originally Posted by 10metreh

The smallest with 1000 digits is 10^999+7 - found in under a second and easily provable.
10^9999+33603 is on the list - I suspect that is the smallest with 10000 digits as I see no other reason for it to be on the list.

Right, that is the smallest 10,000-digit prime. The smallest 100,000-digit prime is 10^99999 + 309403.

[R. Gerbicz]

Originally Posted by CRGreathouse

Right, that is the smallest 10,000-digit prime. The smallest 100,000-digit prime is 10^99999 + 309403.

That is only a prp, see: http://www.primenumbers.net/prptop/prptop.php

What is interesting about this table, that it isn't sorted correctly in decreasing order, there are many errors in the sort. One of them is:

179	10^99999+309403	       100000	Daniel Heuer 01/2004
180	10^100000-260199       100000	Patrick De Geest 01/2010