Thursday, May 10, 2007

Conjectures about 3^n-1 and 3^n-2^n

I am making some educated guesses (conjectures) about 3^n-1 and 3^n-2^2 based on making a table in excel and looking for patterns. When are these numbers prime?

Mathematicians look for patterns and then make conjectures which then they try to prove. So, this is what I am doing:

Conjectures about 3^n-1
1) This result is always even and therefore never prime for n > 1

2) After n=32 the result will always end in zero

3) After n=32 the number of zeros will typically increase so that for large numbers of n there will be a lot of zeros. The ratio of numbers other than zero to zero will decrease as n grows.
For n = 66
result = 30903154382632600000000000000000

For n = 200
result = 265613988875875000000000000000000000000000000000000000000000000000000000000000000000000000000000

Conjectures about 3^n-2^n
1) for n > 31 the result will always end in a zero and not be prime

2) The numbers of zeros is increasing with n increasing

Also, I still need to look at the values left here because they appear prime or possibly prime. I need to write a computer program to check that they're prime. However, all other numbers resulting from 3^n-2^n proved to not be prime.


Alex McFerron said...

It is May 13th and I'm still working on this issue. I started a java programming log where I'll be posting the programming details related to my math questions.
There is a link to it on the right hand side of this blog


Matt Bardoe said...

I don't think your conjecture about 3^n-1 having zeroes can be correct. This would mean that 3^n is congruent to 1 mod 10 for all n>32. But we can find the pattern for 3^n. If n is congruent to 0 mod 4 then 3^n is congruent to 1 mod 10, if n is congruent to 1 mod 4 then 3^n is congruent to 3 mod 10, if n is congruent to 2 mod 4 then 3^n is congruent to 9 mod 10, and if n is congruent to 3 mod 4 then 3^n is congruent to 7 mod 10.

Your error may have come from using Excel which does not handle large arithmetic precisely.

Alex McFerron said...

Thanks for your comment! I'm going to think about what you said and respond in a posting about this topic.