So, we want the vector , from the matrix equation:

where "r" is the row index of the first matrix at left, and "i" his column index.

Note that in the last equation, both r and i start counting from zero for the first row and column.

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P(i) is the partition function

The first few values of the partition function are (starting with p(0)=1):

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, … (sequence A000041 in OEIS; the link has valuable information about the partition function).

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is the number of repetitions of the integer j in the partition of the number i

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Solving the equation

If we do the substitution , we simplify the first equation to:

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Special base.

This equation suggest a special number, which is m=1.7632228343518967102252017769517070804...

m is defined by

For the base a=m, the equation gets simplified to:

But let's forget about m for now.

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We are now very close to the solution. The only obstacle remaining is the product:

If we can do a substitution that get us rid of him, we have the solution:

At this point we only need to substitute , where f is arbitrary, to get:

... and we get:

The choice of f, very probably, determines the value for °a, and the branch of tetration.

where "r" is the row index of the first matrix at left, and "i" his column index.

Note that in the last equation, both r and i start counting from zero for the first row and column.

______________________________________

P(i) is the partition function

The first few values of the partition function are (starting with p(0)=1):

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, … (sequence A000041 in OEIS; the link has valuable information about the partition function).

______________________________________

is the number of repetitions of the integer j in the partition of the number i

______________________________________

Solving the equation

If we do the substitution , we simplify the first equation to:

______________________________________

Special base.

This equation suggest a special number, which is m=1.7632228343518967102252017769517070804...

m is defined by

For the base a=m, the equation gets simplified to:

But let's forget about m for now.

______________________________________

We are now very close to the solution. The only obstacle remaining is the product:

If we can do a substitution that get us rid of him, we have the solution:

At this point we only need to substitute , where f is arbitrary, to get:

... and we get:

The choice of f, very probably, determines the value for °a, and the branch of tetration.

(01/03/2016, 11:24 PM)marraco Wrote:

I have the result, but I do not yet know how to get it.