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The number $K$ is expressible as part of one of either of two quadratic functions with integer coefficients. One of the functions is quadratic in $\pi$ and one is quadratic in $e$:

$$1955e^2+223e-2383=eK$$

$$-134\pi^2+5842\pi-2389=\pi K$$

The sum of the digits of all the coefficients in the first equation is 11. In the second equation this sum is -11. (When summing take into account the sign on the coefficient). Is this just a coincidence or does it point to some special relation?

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Those polynomials are quadratic in $\pi$ and $\mathrm e$, not cubic. –  joriki Sep 19 '12 at 21:27
    
Oops. Duly edited. –  ben Sep 19 '12 at 21:30

2 Answers 2

up vote 1 down vote accepted

The relationship between the sums of the coefficients could be coincidental, as there is no particular reason to write the expression in base ten. But is more likely that somebody looked for this relationship.

As joriki has said, the two numbers are similar up to 15 significant figures.

Make that 17 since the two-digit sum of the digits are also the same apart from sign. But you have used 22 digits in the expressions as well as exercising some other choices, so it is not particularly surprising that such a close relationship can exist.

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The two numbers defined by these two equations are not the same. As calculated by Wolfram|Alpha, the first is $4660.58426632588638\ldots$ whereas the second is $4660.58426632589179\ldots$ .

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Wouldn't we expect $e$ and $\pi$ to be algebraically independent? It is a very close match, though. –  Ross Millikan Sep 19 '12 at 21:56

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