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Statement

Setting (p. 39). The formulas (4) describe $x,y,z$ through pairwise coprime natural numbers $q_0,q_1,\ldots,q_n>1$ and nonzero exponents:

$$x=q_0^{\alpha_0}\prod_{s=1}^nq_s^{\alpha_s},\qquad y=\prod_{s=1}^nq_s^{\beta_s},\qquad z=q_0^{\gamma_0}\prod_{s=1}^nq_s^{\gamma_s},$$

together with the relations $\alpha_0x=\gamma_0z$ and $\alpha_sx+\beta_sy=\gamma_sz$ ($s=1,\ldots,n$), the conditions $(\alpha_0,\gamma_0)=(\alpha_s,\beta_s,\gamma_s)=1$, and the condition printed as $\alpha_{s_i}/\alpha_{s_j}\ne\beta_{s_i}/\beta_{s_j}\ne\gamma_{s_i}/\gamma_{s_j}$, which comes from merging primes whose exponent triples are proportional. (The print lists the factors as $q_0,q_1,\ldots,q_s$, with $s$ for $n$.) The factor $q_0$ thus divides $x$ and $z$ but not $y$.

Lemma 1 (p. 40). If $x,y,z$ defined by the formulas (4) satisfy $x^xy^y=z^z$, then $n>1$.

So the case $n=1$, that is $x=q_0^{\alpha_0}q_1^{\alpha_1}$, $y=q_1^{\beta_1}$, $z=q_0^{\gamma_0}q_1^{\gamma_1}$ (formula (6)), has no solution.

Proof pointer

Pp. 40--43. The paper shows $\max\{\alpha_1,\beta_1\}=\beta_1$, writes $\alpha_0=\gamma_0+a$, $\beta_1=\alpha_1+b$, $\gamma_1=\alpha_1+c$ with $a,b,c>0$, and uses $(\alpha_0,\gamma_0)=1$ to get $\alpha_0=q_1^c$, $\gamma_0=q_0^a$ and $q_1^c-q_0^a=a$. The cases $a=1$, $a=2$ and $a=3$ are excluded by hand through the size of a logarithmic expression. For $a\ge4$, a table of numerical minima excludes $a/q_0^a\ge10^{-3}$; otherwise the inequalities (11) and the results of Baker and Feldman on linear forms in logarithms give (12), $q_0^a,q_1^c<2^{200}$, and further tables computed with 15-digit tables of natural logarithms reduce to $10<a,c<40$ and $32\le q_0,q_1<2^{20}$ (13) and then exclude the remaining values.

Read depth

Claims checked: the setting (4), the statement and the outline of the proof were read clause by clause on the page images of the print. The numerical tables of pp. 41--43 were not recomputed. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204--216; N. I. Feldman, Mat. Zametki 5 (1969), 681--690; tables of natural logarithms (Computing Centre of the USSR Academy of Sciences, 1960).

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 is a step of the paper's proof that every solution of $x^xy^y=z^z$ with $x,y,z>1$ has $x$, $y$, $z$ with the same prime divisors; it does not address whether solutions exist.