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Statement

Setting (pp. 237--238). The equation (1) is $x^xy^y=z^z$ in positive integers. A solution is trivial when $x=1,\ y=z$ or $x=z,\ y=1$. The index of a solution is $Q=xy/z^2$. Mills's Theorem 1, recalled on p. 237, gives no non-trivial solution with $4xy>z^2$, that is $Q>1/4$, and his Theorem 2 gives exactly Ko's family (2) with $4xy=z^2$, that is $Q=1/4$. For the remaining non-trivial solutions, those with $4xy<z^2$, the paper assumes by symmetry $z>x\ge y>1$ (its (3)), so that $Q$ is a rational number with $0<Q<1$.

Here $[t]$ is the greatest integer not exceeding $t$ (p. 238).

Theorem 5 (p. 238). Let $Q=L/R$ with $(L,R)=1$, and let $X=[(\log R)/\log2]$. If $Q\le(X+1)/(X+2)^2$, then the equation $x^xy^y=z^z$ has no non-trivial solutions $x,y,z$ of index $Q$. In particular, if $Q=L/R<1/4$ with $(L,R)=1$ and $1\le L\le5$, the equation has no non-trivial solutions of index $Q$.

Proof pointer

§ 5, pp. 241--243, in the notation of § 1 (pp. 238--239), where $Q=L/R$ with $R=Pr$, $1\le P\le3$ and $r\ge2$, and $p^e\parallel r$, $p^f\parallel a$, $ef\ge1$ for a prime $p$.

• Lemma 6 (p. 241): if $Q\le v/(v+1)^2$ for a real $v>1$, then $(v-1)f<e$; it follows from Lemma 2 of § 3.
• Definition (p. 241): for an integer $w\ge2$, an integer is $w$-free when no $w$-th power of a prime divides it.
• Lemma 7 (p. 241): if $Q\le w/(w+1)^2$ and $r$ is $w$-free, where $w\ge2$ is an integer, then there is no non-trivial solution, since Lemma 6 would give $(w-1)f<e\le w-1$.
• Corollary (pp. 241--242): the first statement of the theorem, because $R$ is $(X+1)$-free.
• Lemmas 8--11 (pp. 242--243) give the second statement for $L=1,\ldots,5$: $Q=1/R$ with $R\ge5$; $Q=2/R$ with $(R,2)=1$, $R\ge9$; $Q=3/R$ with $(R,3)=1$, $R\ge13$; $Q=4/R$ with $(R,2)=1$, $R\ge17$; and $Q=5/R$ with $(R,5)=1$, $R\ge21$. Lemmas 8--10 are proved by Lemma 7, with Theorem 4 for $R=9$ and $R=16$ and a direct computation for $Q=3/13$. Lemma 11 ($L=4$ and $L=5$) is introduced by the words "In quite a similar manner we can prove" (p. 243); its proof is not written out.

Read depth

Claims checked: the statement, Lemmas 6--10, the Corollary and their proofs on pp. 241--243 were read clause by clause on the page images of the print. The second statement for $L=4$ and $L=5$ rests on Lemma 11, whose proof the paper omits. The facts of § 1 taken from Mills and Schinzel are cited, not proved, in the paper and were not read. Nothing here is independently reviewed.

Dependencies

Theorem 4 (used in Lemmas 9 and 10). External inputs named by the paper: Mills's 1959 report and Schinzel (1958) for the facts of § 1, and Dem'janenko (1975) through Lemma 2 of § 3.

Source. S. Uchiyama, On the Diophantine equation $x^xy^y=z^z$, Trudy Mat. Inst. Steklov. 163 (1984), 237--243; the edition read is named on the source card.

Bears on

• Problem 674: the theorem does not touch the problem's question, which the family (2) that the paper recalls from Ko already answers. It excludes non-trivial solutions with $4xy<z^2$ for the indices it names.