PARI's commands

[CRGreathouse]

So what are 3*1, 3*4, and 3*7 mod 9? You're almost there.

[science_man_88]

Originally Posted by CRGreathouse

So what are 3*1, 3*4, and 3*7 mod 9? You're almost there.

they all come back to 3 so assuming my idea is correct the first column must always be 3 hence we can map it through the diagonals and get the modulo for the Mersenne primes to come.

though why multiply them in the next in the row they cover almost all the same range.

[science_man_88]

the hard part is how to map it accurately without knowing it.

[CRGreathouse]

Originally Posted by science_man_88

they all come back to 3 so assuming my idea is correct the first column must always be 3

Right. This is a proof that the first column is always 3 where it exists. (We don't know if there are infinitely many rows because we don't know if there are infinitely many Mersenne primes.)

Originally Posted by science_man_88

hence we can map it through the diagonals and get the modulo for the Mersenne primes to come.

You can't tell anything about future Mersenne primes because no matter what they are, you'll get a 3. Right?

[science_man_88]

Originally Posted by CRGreathouse

Ah, I think I have it. Together with my above code take

ProdK(k)=k--;vector(#A43-k,i,prod(j=0,k,drMersenne(A43[i+j]))%9)

which gives the digital roots of the product of k consecutive Mersenne numbers. So ProdK(1) gives A135928(n), ProdK(2) gives A010888(A165223(n)), etc.

Then you have

you say Mersenne numbers ... any other patterns in it if it is for Mersenne primes such that it would help predict modulo ?

[science_man_88]

can this be turned into a triangle by turning it 45 degrees ?

I see more weird things in the diagonals that would become rows of this triangle not saying they tell much lol.

[CRGreathouse]

Originally Posted by science_man_88

you say Mersenne numbers ... any other patterns in it if it is for Mersenne primes such that it would help predict modulo ?

I'm sorry, I meant Mersenne primes. The same pattern would hold for Mersenne numbers with exponents relatively prime to 6.

You *know* that you can't predict anything about any sequence that has exponents that are relatively prime to 6, because you're already shown how those sequences have to start. It cuts both ways: because the proof shows that all such sequences will have the same behavior, you can't use that behavior to predict anything within those sequences.

[CRGreathouse]

Originally Posted by science_man_88

can this be turned into a triangle by turning it 45 degrees ?

Sure, whatever you want. Feel free to modify my code as needed.

[science_man_88]

>for(k=1,15,print(k"\t"ProdK(k)))
1       [3, 7, 4, 1, 1, 4, 1, 1, 1, 4, 4, 1, 4, 1, 1, 1, 1, 1, 4, 1, 4, 4, 4, 4, 4, 1, 1, 4, 1, 1, 1, 4, 4, 1, 4, 4, 4, 4, 1]
2       [3, 1, 4, 1, 4, 4, 1, 1, 4, 7, 4, 4, 4, 1, 1, 1, 1, 4, 4, 4, 7, 7, 7, 7, 4, 1, 4, 4, 1, 1, 4, 7, 4, 4, 7, 7, 7, 4]
3       [3, 1, 4, 4, 4, 4, 1, 4, 7, 7, 7, 4, 4, 1, 1, 1, 4, 4, 7, 7, 1, 1, 1, 7, 4, 4, 4, 4, 1, 4, 7, 7, 7, 7, 1, 1, 7]
4       [3, 1, 7, 4, 4, 4, 4, 7, 7, 1, 7, 4, 4, 1, 1, 4, 4, 7, 1, 1, 4, 4, 1, 7, 7, 4, 4, 4, 4, 7, 7, 1, 1, 1, 4, 1]
5       [3, 4, 7, 4, 4, 7, 7, 7, 1, 1, 7, 4, 4, 1, 4, 4, 7, 1, 4, 4, 7, 4, 1, 1, 7, 4, 4, 7, 7, 7, 1, 4, 4, 4, 4]
6       [3, 4, 7, 4, 7, 1, 7, 1, 1, 1, 7, 4, 4, 4, 4, 7, 1, 4, 7, 7, 7, 4, 4, 1, 7, 4, 7, 1, 7, 1, 4, 7, 7, 4]
7       [3, 4, 7, 7, 1, 1, 1, 1, 1, 1, 7, 4, 7, 4, 7, 1, 4, 7, 1, 7, 7, 7, 4, 1, 7, 7, 1, 1, 1, 4, 7, 1, 7]
8       [3, 4, 1, 1, 1, 4, 1, 1, 1, 1, 7, 7, 7, 7, 1, 4, 7, 1, 1, 7, 1, 7, 4, 1, 1, 1, 1, 4, 4, 7, 1, 1]
9       [3, 7, 4, 1, 4, 4, 1, 1, 1, 1, 1, 7, 1, 1, 4, 7, 1, 1, 1, 1, 1, 7, 4, 4, 4, 1, 4, 7, 7, 1, 1]
10      [3, 1, 4, 4, 4, 4, 1, 1, 1, 4, 1, 1, 4, 4, 7, 1, 1, 1, 4, 1, 1, 7, 7, 7, 4, 4, 7, 1, 1, 1]
11      [3, 1, 7, 4, 4, 4, 1, 1, 4, 4, 4, 4, 7, 7, 1, 1, 1, 4, 4, 1, 1, 1, 1, 7, 7, 7, 1, 4, 1]
12      [3, 4, 7, 4, 4, 4, 1, 4, 4, 7, 7, 7, 1, 1, 1, 1, 4, 4, 4, 1, 4, 4, 1, 1, 1, 1, 4, 4]
13      [3, 4, 7, 4, 4, 4, 4, 4, 7, 1, 1, 1, 4, 1, 1, 4, 4, 4, 4, 4, 7, 4, 4, 4, 4, 4, 4]
14      [3, 4, 7, 4, 4, 7, 4, 7, 1, 4, 4, 4, 4, 1, 4, 4, 4, 4, 7, 7, 7, 7, 7, 7, 7, 4]
15      [3, 4, 7, 4, 7, 7, 7, 1, 4, 7, 7, 4, 4, 4, 4, 4, 4, 7, 1, 7, 1, 1, 1, 1, 7]

here's one such thing I see.

[CRGreathouse]

SLSN, for that one. Since you're going along the antidiagonals there aren't any more terms to check...

[science_man_88]

Originally Posted by CRGreathouse

SLSN, for that one. Since you're going along the antidiagonals there aren't any more terms to check...

SLSN ?