## 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:

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

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

cheers,
alex

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...