It is referenced that in

Chowla, S., Proof of a conjecture of Julia Robinson, Norske Vid. Selsk. Forh. (Trondheim) 34, 100–101 (1961),

it is shown that the equation $\epsilon_1 + \epsilon_2 = 1$ has only finitely many solutions, when $\epsilon_1, \epsilon_2$ are units of a given number field. Unfortunately, the article seems to be unavailable electronically. I hence wonder if the argument given in the article is available from another source or if it could be reproduced here, if not too lengthy.

Thank you.


2 Answers 2


This is a standard result, and is true for $S$-units. It is proven in many texts, although sometimes the reader is referred elsewhere for the estimate from Diophantine approximation that is needed (such as Thue's or Roth's theorem). The full proof is contained in my book with Hindry, Diophantine Geometry, GTM 201, Springer, Section D.8 (Application: The Unit Equation $U+V=1$). The reduction of the unit equation to Diophantine approximation is in my elliptic curves book (and in many many other books). There are also quantitative results (upper bounds for the number of solutions) and effective results (upper bounds for the largest solution, using linear forms in logs).


For what it is worth, I reproduce below the relevant paragraphs in Chowla's aforementioned paper.

The number of solutions of the equation

$$\epsilon - \epsilon^{\prime} =1 \qquad \qquad \qquad(1)$$

where $\epsilon$ and $\epsilon^{\prime}$ are units of a fixed algebraic number field $R(\theta)$ is finite.

Proof. Let $n$ be the degree of the field $R(\theta)$. With the usual notation we write $n=r_{1}+2r_{2}$, $r=r_{1}+r_{2}-1$ (see e.g. the Chelsea reprint of Landau's Algebraische Zahlen) and (in accordance with the Dirichlet-Minkowski theorem)

$$\epsilon = \rho\eta_{1}^{a_{1}} \cdots \eta_{r}^{a_{r}}, \quad \epsilon^{\prime} = \rho^{\prime} \eta_{1}^{a_{1}^{\prime}} \cdots \eta_{r}^{a_{r}^{\prime}}$$

where $\eta_{1}, \ldots, \eta_{r}$ is a system of fundamental units of $R(\theta)$ and $\rho, \rho^{\prime}$ are certain roots of unity belonging to $R(\theta)$. Write for $m\in\{1,\ldots, r\}$

$$a_{m} = q_{m}(2n+1)+t_{m}, \quad a_{m}^{\prime} = q_{m}^{\prime}(2n+1)+t_{m}^{\prime}$$

where the $q$'s and $t$'s are rational integers and $0\leq t_{m}, t_{m}^{\prime} \leq 2n$. Then $(1)$ becomes

$$\rho \, \eta_{t_{1}}\cdots \eta_{r}^{t_{r}} \alpha^{2n+1}-\rho^{\prime} \, \eta_{1}^{t_{1}^{\prime}} \cdots \eta_{r}^{t_{r}^{\prime}} \beta^{2n+1}=1$$

where $\alpha, \beta$ are integers (in fact, units) of $R(\theta)$. By a well-known extension of the Thue-Siegel-Roth theorem (see, for example, vol. 2 of LeVeque's "Topics in number theory", pp. 150-154) the equation

$$\lambda \alpha^{2n+1} - \mu\beta^{2n+1} = 1$$

where $\lambda, \mu$ are fixed integers of $R(\theta)$ and $\alpha, \beta$ are the unknowns (also integers of $R(\theta)$) has only a finite numbers of solutions. Hence our assertion regarding $(1)$.

[Note. $\rho$ and $\rho^{\prime}$ can assume only a finite number of values.]


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