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Statement

Setting (pp. 39--40 and 44). A solution of $x^xy^y=z^z$ is written in the shape (4) (see Lemma 1) with pairwise coprime $q_0,q_1,\ldots,q_n>1$, where $q_0$ divides $x$ and $z$ but not $y$. Since $\min\{\alpha_s,\beta_s\}<\gamma_s\le\max\{\alpha_s,\beta_s\}$, the proof (p. 44) orders the indices so that, with integers $a_i,b_j,c_i,c_j\ge0$,

$$\alpha_0=\gamma_0+a,\qquad \alpha_i=\beta_i+a_i,\quad \gamma_i=\beta_i+c_i\quad(i=1,\ldots,t),$$ $$\beta_j=\alpha_j+b_j,\quad \gamma_j=\alpha_j+c_j\quad(j=t+1,\ldots,n),$$

and sets (16)

$$a_i'=a_i-c_i,\qquad b_j'=\frac{c_j\gamma_0}{a}+c_j-b_j,\qquad c_i'=c_i-\frac{a_i-c_i}{a}\gamma_0 .$$

Lemma 2 (pp. 43--44). If $x,y,z$ do not have the same prime divisors, then

$$x=A^mB^n,\qquad y=B^{n-1},\qquad z=A^{m-1}B^n,$$

where

$$A=\frac{m-1}{m},\qquad B=\prod_{i=1}^tq_i^{c_i'}\prod_{j=t+1}^nq_j^{b_j'},\qquad m=\frac{\alpha_0}{a},\qquad n=\frac{\alpha_0^{\alpha_0/a}}{\alpha_0^{\alpha_0/a}-Ba\gamma_0^{\gamma_0/a}},$$

and (14)

$$\alpha_0=\prod_{j=t+1}^nq_j^{c_j},\qquad \gamma_0=q_0^a\prod_{i=1}^tq_i^{a_i-c_i},\qquad \alpha_0-\gamma_0=a.$$

The print uses the letter $n$ both for the number of factors $q_1,\ldots,q_n$ and for the exponent defined in the lemma.

Proof pointer

P. 44. Solving the relations of (4) for $\gamma_0$, $\beta_i$ and $\alpha_j$ gives the system (15); the coprimality $(\alpha_0,\gamma_0)=1$ then yields the expressions for $\alpha_0$ and $\gamma_0$ in (14). With the quantities (16) and $D=\alpha_0^{\alpha_0/a}-Ba\gamma_0^{\gamma_0/a}$ the paper writes every exponent of (4) in the form (17), and substituting (17) into (4) gives (14).

Read depth

Claims checked: the statement, the notation it draws from the proof, and the outline of the proof were read clause by clause on the page images of the print. The algebra from (15) to (17) was not rederived. Nothing here is independently reviewed.

Dependencies

None.

Source. V. A. Demʹjanenko, On a conjecture of A. Schinzel, Izv. Vysš. Učebn. Zaved. Matematika 1975, no. 8 (159), 39--45; the edition read is named on the source card.

Bears on

• Problem 674: the lemma describes the form a solution of $x^xy^y=z^z$ with $x,y,z>1$ would take if $x$, $y$, $z$ did not have the same prime divisors, as a step of the paper's proof that this does not happen; it does not address whether solutions exist.