Pie?

π\pi and ee are both transcendental numbers. Transcendental numbers are numbers which can't be roots of a non-zero polynomial equation with rational coefficients. Keeping that in mind, read the following statements.

[1][1]. It is possible for π+e\pi+e to be a rational number in base 1010.

[2][2]. At least one of π+e\pi+e and π×e\pi \times e is irrational in base 1010.

[3][3]. π\pi is an irrational number in base 1010.

Which of these statements are correct?

Note: This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.

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