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

If $n^x$ is an integer for every positive integer $n$, then $x$ must be an integer; this is a well-known exercise. If only $2^x$ and $3^x$ are known to be integers, the same conclusion is expected, but nobody can prove it: it would follow from the four exponentials conjecture of transcendental number theory. With $2$, $3$ and $5$ in place of $2$ and $3$, it is a theorem.

Problem

Let $x$ be a real number such that $2^x$ and $3^x$ are both integers. Must $x$ be an integer?

Since $2^x$ is then a positive integer, $x \ge 0$; and if $x = p/q$ is rational, then $2^p = (2^x)^q$ forces $q$ to divide $p$, so $x$ is an integer. The whole difficulty is to rule out irrational $x$.

Background

Alaoglu and Erdős (1944), studying colossally abundant numbers, remarked that $p^x$ and $q^x$ very likely cannot both be rational for distinct primes $p$, $q$ unless $x$ is an integer, and reported Siegel's result that this holds for three primes. The question was asked on MathOverflow by Alon Amit in 2010; Serge Lang is reported there to have called the $2^x, 3^x$ case a notorious open problem. Kevin Buzzard's answer there relates it to the arithmetic of Grössencharacters, where the analogous statement for all $n$ is a theorem.

What is known

• The four exponentials conjecture says: if $x_1, x_2$ are linearly independent over $\mathbb{Q}$ and $y_1, y_2$ are linearly independent over $\mathbb{Q}$, then at least one of $e^{x_i y_j}$ ($i, j \in \{1, 2\}$) is transcendental. With $x_1 = 1$, $x_2 = x$, $y_1 = \log 2$, $y_2 = \log 3$ it would show that $x$ is rational, hence an integer.
• The six exponentials theorem of Lang and Ramachandra (1960s), the same statement with three numbers $y_j$, proves the result when $2^x$, $3^x$ and $5^x$ are all integers. An elementary proof of that case is due to Waldschmidt.
• If $n^x$ is an integer for every positive integer $n$, then $x$ is an integer; this appeared on the 1971 Putnam competition.

What would count as progress

• A proof that $2^x, 3^x \in \mathbb{Z}$ forces $x \in \mathbb{Z}$, or the same for any single fixed pair of primes.
• Any new case of the four exponentials conjecture, for instance for pairs $y_1, y_2$ of logarithms of integers, which is exactly what this question needs.
• A formal proof of one of the known variants in Mathoverflow/17560.lean (the case of all $n$, or of $2$, $3$ and $5$), which would be a step towards formal transcendence theory; the main statement there is mathoverflow_17560.

References

1. L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc. 56 (1944), 448–469.
2. K. Ramachandra, Contributions to the theory of transcendental numbers I, Acta Arith. 14 (1968), 65–72.
3. A. Amit and others, If 2^x and 3^x are integers, must x be as well?, MathOverflow question 17560 (2010).

Source

Imported from Formal Conjectures Mathoverflow/17560, curated by Google DeepMind and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.