Uchiyama 1984 diophantine equation

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theorem_3: Uchiyama's theorem that for each fixed value Q < 1/4 of the index xy/z^2 the equation x^x y^y = z^z has at most finitely many non-trivial solutions, and that all of them can be determined effectively.

theorem_4: Uchiyama's theorem that the equation x^x y^y = z^z has no non-trivial solution whose index xy/z^2 equals (1-k^2)/4 for a rational k with 0 < k < 1.

theorem_5: Uchiyama's theorem that x^x y^y = z^z has no non-trivial solution whose index Q = L/R in lowest terms satisfies Q <= (X+1)/(X+2)^2 with X = [log R/log 2], and in particular none with Q < 1/4 and 1 <= L <= 5.

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Uchiyama, S., On the {D}iophantine equation {$x^x y^y=z^z$}. Trudy Mat. Inst. Steklov. 163 (1984), 237--243.

The paper surveys and extends work on x^x y^y = z^z (p. 237). It recalls that Erdos conjectured, as the paper states it, that the equation has no nontrivial positive-integer solutions, a solution being trivial when x = 1, y = z or x = z, y = 1; that Ko Chao proved there are none with gcd(x,y) = 1 and gave the infinite family (2) of nontrivial solutions with gcd(x,y) > 1; that Mills's Theorems 1 and 2 show there are no nontrivial solutions with 4xy > z^2 and only the family (2) with 4xy = z^2, together with Mills's remark that there are none with 4xy < z^2 and (x,y) < 6^150; that Schinzel proved that if x, y, z are all different from 1 then every prime factor of x divides y or every prime factor of y divides x; and that Dem'janenko proved Schinzel's conjecture that x, y, z > 1 in a solution have the same prime factors, by means of the Baker--Fel'dman inequality for linear forms in logarithms of algebraic numbers. Uchiyama works in the remaining range 4xy < z^2, assuming z > x >= y > 1 and parametrizing a solution by its index Q = xy/z^2, which then lies in (0,1/4) (p. 238). Theorem 3 states that for each fixed Q < 1/4 the equation has at most finitely many nontrivial solutions and that all of them can be determined effectively; Theorem 4 rules out solutions when Q = (1-k^2)/4 with k rational, 0 < k < 1; Theorem 5 rules them out when Q = L/R in lowest terms with Q <= (X+1)/(X+2)^2 where X = [(log R)/log 2], and in particular for Q < 1/4 with 1 <= L <= 5. For L = 4 and L = 5 that last statement rests on Lemma 11 (p. 243), whose proof the paper does not write out. The proofs are divisibility and inequality arguments on the exponents in Mills's notation (pp. 238--239), together with Dem'janenko's theorem, which enters through Lemma 1 (p. 239) and Lemma 2 (p. 240). The paper also gives a new proof of Mills's Theorem 1 (pp. 239--240).

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Bears on.

• Problem 674: the paper does not settle the problem's question; the family (2) it recalls from Ko (p. 237) gives solutions with x, y, z > 1. Its own Theorems 3--5 concern the nontrivial solutions with 4xy < z^2, the only ones outside the family (2) that Mills's theorems leave possible: finitely many for each index below 1/4, and none for the indices Theorems 4 and 5 name. They do not decide whether any such solution exists.

Results.

• Theorem 3 (p. 238): for each fixed index Q < 1/4 the equation has at most finitely many nontrivial solutions, all effectively determinable.
• Theorem 4 (p. 238): no nontrivial solution has index Q = (1-k^2)/4 with k rational, 0 < k < 1.
• Theorem 5 (p. 238): no nontrivial solution has index Q = L/R, (L,R) = 1, with Q <= (X+1)/(X+2)^2 and X = [(log R)/log 2]; in particular none with Q < 1/4 and 1 <= L <= 5.

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