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I am experimenting with a solver for finding algebraic dependencies and would like to test it on more data sets.

Are there transcendental numbers that are "suspected" to be algebraically dependent without conjectured relation?

Note that conjectured equality is not an answer because the relation is conjectured.

Especially interested in zeta at odd integers (no matter if it is transcendental or not) and other set of transcendental numbers.

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An interesting 1897 conjecture by the Indiana state legislature suggests that you test your software with the pair $(\pi,32)$ –  Ben Crowell Oct 13 '12 at 15:50
    

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