Demjanenko 1975 conjecture

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lemma_1: Demʹjanenko's Lemma 1: if x, y, z given by the formulas (4) satisfy x^x y^y = z^z, then the number n of factors q_1, ..., q_n in (4) exceeds 1.

lemma_2: Demʹjanenko's Lemma 2: if a solution x, y, z of x^x y^y = z^z in the shape (4) does not have the same prime divisors, then x = A^m B^n, y = B^(n-1), z = A^(m-1) B^n with A, B, m, n and the exponents of q_0 given by (14).

main_theorem: Demʹjanenko's theorem, stated on p. 39 as a proof of Schinzel's 1958 conjecture, that natural numbers x, y, z different from 1 satisfying x^x y^y = z^z have the same prime divisors.

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Demʹjanenko, V. A., On a conjecture of A. Schinzel. Izv. Vysš. Učebn. Zaved. Matematika 1975, no. 8 (159), 39--45.

This Russian-language note addresses Schinzel's 1958 conjecture that natural numbers x, y, z, each different from 1, satisfying x^x y^y = z^z must have the same prime divisors; the author says that, as far as he knows, it had not been proved, and the note presents a proof (p. 39). Assuming a counterexample, he writes x, y, z as products over a common set of primes p_i (formula (2)) and derives the linear relations a_i x + b_i y = c_i z (formula (3)), reducing to the shape (4) in which x, y, z are built from pairwise coprime q_0,...,q_n > 1 with q_0 dividing x and z but not y. He shows z < x+y, since z >= x+y would give 0 = ln(z^z/x^x y^y) >= x ln(1+y/x) + y ln(1+x/y) > 0; with d = (x,y) this yields the identity (5), from which he derives min{alpha_s,beta_s} < gamma_s <= max{alpha_s,beta_s}. Lemma 1 (p. 40) rules out 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}, by a case analysis on a = alpha_0 - gamma_0 (the cases a = 1, 2, 3 by hand; for a >= 4, tables of numerical minima together with the bound q_0^a, q_1^c < 2^200 of (12), obtained from the results of Baker and Feldman on linear forms in logarithms). Lemma 2 (pp. 43--44) gives an explicit parametrization (14) of a solution whose x, y, z do not have the same prime divisors; the same results of Baker and Feldman then bound alpha_0, gamma_0 < 2^200 in (19), and the author states that the method of Lemma 1 excludes the remaining values (p. 45). Problem 674 lists the paper among its references.

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Bears on. #674: the problem asks whether x^x y^y = z^z has integer solutions with x, y, z > 1. The paper does not address whether solutions exist; its main theorem (p. 39) states that in every such solution x, y and z have the same prime divisors.

Results.

• Main theorem (p. 39): natural numbers $x,y,z$ different from $1$ with $x^xy^y=z^z$ have the same prime divisors (Schinzel's conjecture).
• Lemma 1 (p. 40): if $x,y,z$ given by the formulas (4) satisfy $x^xy^y=z^z$, then $n>1$.
• Lemma 2 (pp. 43--44): if $x,y,z$ do not have the same prime divisors, then $x=A^mB^n$, $y=B^{n-1}$, $z=A^{m-1}B^n$ with $A$, $B$, $m$, $n$ as defined in the lemma and $\alpha_0$, $\gamma_0$ as in (14).

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