Must x be an integer if 2^x and 3^x are integers?

Show that a real number x for which both 2^x and 3^x are integers must itself be an integer.

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	1	+{
	2	+ "domain": "mathematics",
	3	+ "topics": [
	4	+ "number-theory"
	5	+ ],
	6	+ "type": "proof",
	7	+ "cover": {
	8	+ "motif": "blocks"
	9	+ },
	10	+ "source": {
	11	+ "key": "formal-conjectures",
	12	+ "name": "Formal Conjectures",
	13	+ "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Mathoverflow/17560.lean",
	14	+ "id": "Mathoverflow/17560",
	15	+ "curator": "Google DeepMind and contributors",
	16	+ "poser": "L. Alaoglu and P. Erdős (1944); asked on MathOverflow by Alon Amit",
	17	+ "status": "open",
	18	+ "as_of": "2026-10-20",
	19	+ "catalog": "0521c19bbd38"
	20	+ }
	21	+}