Tengely 2020 diophantine equation erdos graham

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conjecture_3_7: Tengely, Ulas and Zygadło's conjecture, from their greedy computations, that every solution of n/2^n = sum of a_i/2^(a_i) has a_k between k + n and 2(k + n), with Remark 3.8 proving the upper bound when n >= 2^k - k.

corollary_2_9: Tengely, Ulas and Zygadło's bound on the largest term of a k-term solution of n/2^n = sum of a_i/2^(a_i) in terms of k alone, which leaves finitely many, effectively computable solutions for each fixed k.

corollary_3_6: Tengely, Ulas and Zygadło's infinite family of rationals x each having at least nine representations as an infinite sum of a_i/2^(a_i) over a strictly increasing sequence of positive integers.

proposition_3_1: Tengely, Ulas and Zygadło's explicit arithmetic progression of values k for each of which n/2^n = sum of a_i/2^(a_i) has at least five solutions with k terms, found by solving discrete logarithm problems.

theorem_2_1: Tengely, Ulas and Zygadło's necessary conditions on a k-term solution of n/2^n = sum of a_i/2^(a_i): n is at most 2^(k+1) - k - 2, a_1 lies between n + 1 and n + 3, 2^(a_k - a_(k-1)) divides a_k, and the first j terms are n + 1, ..., n + j once n is at least 2^(j+1) - j.

theorem_2_5: Tengely, Ulas and Zygadło's computer enumeration of every solution of n/2^n = sum of a_i/2^(a_i) with k terms for each k from 2 to 8, among them n = 1 with k = 3 and k = 7.

theorem_2_8: Tengely, Ulas and Zygadło's bound a_k <= 2n + 2k log_2 k on the largest term of any solution of n/2^n = sum of a_i/2^(a_i) with k terms.

theorem_3_5: Tengely, Ulas and Zygadło's computer verification that for each n with 2 <= n <= 10^4 the equation n/2^n = sum of a_i/2^(a_i) has a solution in k, a_1, ..., a_k with first term a_1 = n + 1, found by a modified greedy algorithm.

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Tengely, Szabolcs and Ulas, Maciej and Zygadło, Jakub, On a Diophantine equation of Erdős and Graham. J. Number Theory 217 (2020), 445--459. https://doi.org/10.1016/j.jnt.2020.05.006

The copy read for this card is the arXiv preprint arXiv:2008.01501v1 (13 pp.); the theorem numbers and pages below are those of that copy.

The paper studies the polynomial-exponential Diophantine equation (1), n/2^n = sum_{i=1}^{k} a_i / 2^{a_i} with k > 1 and a_1 < ... < a_k, posed in the Erdos-Graham monograph (p. 63) and previously attacked by Borwein and Loring. Theorem 2.1 (p. 2) shows that if a solution exists for fixed k then n <= 2^{k+1}-k-2, that n + 1 <= a_1 <= n + 3 and 2^{a_k - a_{k-1}} divides a_k, and that a_i = n + i for i = 1, ..., j when n >= 2^{j+1} - j for some 1 <= j < k. Theorem 2.5 (p. 5) enumerates all solutions for 2 <= k <= 8. Theorem 2.8 (p. 6) proves a_k <= 2n + 2k log_2 k, and Corollary 2.9 (p. 6) deduces a_k <= 2^{k+2} + 2k(log_2 k - 1) - 4, which the paper presents as the affirmative answer to its Question 1.2 (p. 2): for each fixed k there are only finitely many solutions, and they can be computed effectively. Proposition 3.1 (p. 6), proved by solving discrete logarithm problems, gives an arithmetic progression of values k for each of which the equation has five or more solutions; its proof text names a different set of four auxiliary parameters from the one the printed progression satisfies, which the result page records. Using a modified greedy algorithm, Theorem 3.5 (p. 8) verifies that every n with 2 <= n <= 10^4 admits a solution with a_1 = n + 1 (the print has "a_i = n+1", read as a_1), extending Borwein and Loring's check of n <= 10^3; n = 1 has solutions in Theorem 2.5. Corollary 3.6 (p. 10) gives infinitely many rationals each with at least nine representations as an infinite sum of terms a_i/2^{a_i}, improving the three of Corollary 2.6 (p. 5). Conjecture 3.7 (p. 11) predicts n + k <= a_k <= 2(n + k), and Remark 3.8 (p. 12) proves the upper bound when n >= 2^k - k.

Source: https://arxiv.org/abs/2008.01501. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2008.01501), every other right reserved.

Read status. Claims checked: Theorems 2.1, 2.5, 2.8 and 3.5, Corollaries 2.9 and 3.6, Proposition 3.1 and Conjecture 3.7 with Remark 3.8 were read clause by clause on the page images on 2026-10-08, for the result pages listed below. The solutions listed in Theorem 2.5 and the residue class of Proposition 3.1 were recomputed in exact arithmetic, as each page records; the paper's searches and greedy computations were not rerun, and the proofs were read but not checked step by step. The claim page Tengely, Ulas and Zygadło 2020 consumes Theorems 2.1, 2.5 and 3.5, Corollaries 2.9 and 3.6, Proposition 3.1 and Conjecture 3.7 with Remark 3.8.

Bears on. #261, whose finite sums with at least two distinct terms are the solutions of (1): Theorem 3.5, with n = 1 from Theorem 2.5, verifies the problem's second question, whether every n has the property, for each n <= 10^4, and leaves it open beyond; Theorem 2.1, Theorem 2.8 and Corollary 2.9 bound the solutions with a fixed number of terms, finitely many for each, which settles neither the first nor the second question; Remark 2.2 records Borwein and Loring's identity, a solution for n = 2^{k+1} - k - 2 with each k > 1, which gives the first question's infinitely many n and is their result, not this paper's; for the third question, a rational with 2^{aleph_0} representations, Corollary 3.6 gives infinitely many rationals with at least nine representations and does not decide it.

Results.

• Theorem 2.1 (p. 2): necessary conditions on a solution.
• Theorem 2.5 (p. 5): all solutions for 2 <= k <= 8, with Theorem 2.3, Corollary 2.4 and Corollary 2.6 summarized there.
• Theorem 2.8 (p. 6): a_k <= 2n + 2k log_2 k.
• Corollary 2.9 (p. 6): a_k bounded in terms of k alone.
• Proposition 3.1 (p. 6): a residue class of k with at least five solutions, with Conjectures 3.2 and 3.3.
• Theorem 3.5 (p. 8): every 2 <= n <= 10^4 has a solution.
• Corollary 3.6 (p. 10): infinitely many rationals with at least nine representations.
• Conjecture 3.7 (p. 11): n + k <= a_k <= 2(n + k), with Remark 3.8 (p. 12).

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