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Statement

The paper prints no numbered theorem. On p. 39 it recalls Schinzel's conjecture of 1958, citing Sierpiński's book (its reference [1]), and announces that the note proves it.

Main theorem (p. 39, Schinzel's conjecture as the paper states it). If natural numbers $x,y,z$, each different from $1$, satisfy

$$x^xy^y=z^z,$$

then $x$, $y$ and $z$ have the same prime divisors.

The paper says (p. 39) that, as far as the author knows, the conjecture had not been proved before.

Proof pointer

Pp. 39--45. Suppose a solution of (1) in which $x,y,z$ do not consist of the same primes. Writing $x,y,z$ over a common list of primes (formula (2)) gives the linear relations $a_ix+b_iy=c_iz$ of (3), and grouping primes with proportional exponents reduces the solution to the shape (4): pairwise coprime natural numbers $q_0,q_1,\ldots,q_n>1$ with $x=q_0^{\alpha_0}\prod_{s=1}^nq_s^{\alpha_s}$, $y=\prod_{s=1}^nq_s^{\beta_s}$, $z=q_0^{\gamma_0}\prod_{s=1}^nq_s^{\gamma_s}$ and $\alpha_0x=\gamma_0z$, $\alpha_sx+\beta_sy=\gamma_sz$. Every solution of (1) has $z<x+y$ (p. 39), since $z\ge x+y$ would make $0=\ln(z^z/x^xy^y)\ge x\ln(1+y/x)+y\ln(1+x/y)>0$. With $d=(x,y)$ this gives the identity (5), and from it the paper derives $\min\{\alpha_s,\beta_s\}<\gamma_s\le\max\{\alpha_s,\beta_s\}$ (pp. 39--40). Lemma 1 (p. 40) excludes the case $n=1$, and Lemma 2 (pp. 43--44) parametrizes the general case as (14). The paper then bounds $0<B<3\gamma_0$ (p. 45), arrives at the system (18), and from the results of Baker and Feldman on linear forms in logarithms (its references [2], [3]) obtains (19), $\alpha_0,\gamma_0<2^{200}$. The last sentence of the paper says that the method of Lemma 1 shows that no $\alpha_0,\gamma_0$ satisfy (18) and (19); the paper prints no computation for this step.

Read depth

Claims checked: the statement, the setup (2)--(5), the inequality $z<x+y$, both lemmas and the closing steps (14)--(19) were read clause by clause on the page images of the print. The numerical tables of pp. 41--43 and the final exclusion step were not checked. Nothing here is independently reviewed.

Dependencies

Lemma 1 and Lemma 2 of the same paper. 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, An inequality for a linear form in logarithms of algebraic numbers, Mat. Zametki 5 (1969), 681--690; and 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 problem asks whether $x^xy^y=z^z$ has integer solutions with $x,y,z>1$. The paper does not address whether solutions exist; its theorem concerns every solution with $x,y,z>1$ and states that $x$, $y$ and $z$ then have the same prime divisors.