Bajpai 2024 effective unit equations beyond three terms

../

theorem_1: Over a number field K with a set S of at most three places containing the infinite ones, the nondegenerate solutions in S-units of a fixed five-term equation a_1u_1 + ... + a_5u_5 = 0 have effectively computable bounded height.

theorem_10: If two distinct representations of N as 2^a 3^b + 2^c + 3^d give an equation with a vanishing subsum, then N is special of type I, II or III, and for each type the paper gives explicit bounds and extremal values of omega(N).

theorem_11: If omega(N) >= 3 and N is not special of type I, II or III, then N is one of 274, 473, 505, 1109, 1595, 1811, 2297, 2779, 4403 and 20761, and each of these has omega(N) = 3.

theorem_3: The number omega(N) of representations of N as 2^a 3^b + 2^c + 3^d, counted by their sets of three summands, is at most 9 for every positive N and at most 8, 7, 6, 5, 4 from N >= 300, 786, 2316, 19700, 131082 on, with the extremal N listed.

theorem_6: The effective height bound of Theorem 1 persists when the five coefficients vary with the solution, provided each has height at most kappa_1 times h(u)^kappa_2; the bound depends only on K, S, kappa_1 and kappa_2.

theorem_8: There are exactly 1431 primitive solutions of u_1 + ... + u_5 = 0 in integers whose prime factors are at most 3 and with no vanishing subsums, and the largest leading term is 3^12 = 531441.

---

Bajpai, Prajeet and Bennett, Michael A., Effective {$S$}-unit equations beyond three terms: {N}ewman's conjecture. Acta Arith. 214 (2024), 421--458, DOI 10.4064/aa230725-14-9. The copy read for this card is arXiv:2308.05162v1, submitted 9 August 2023, whose first page dates the manuscript August 11, 2023; 30 pages. Labels and pages on this card and its result pages are that version's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2308.05162), every other right reserved.

Theorem 1 (p. 2) gives an effectively computable upper bound for the heights of the nondegenerate solutions in S-units of a fixed five-term equation a_1u_1 + ... + a_5u_5 = 0 over a number field K, when the set S of places contains the infinite places and has at most three elements; a solution is degenerate when a_iu_i + a_ju_j = 0 for some i < j. This extends Vojta's effective treatment of four terms with |S| <= 3 (p. 1). Lower bounds for linear forms in complex and p-adic logarithms make the three largest terms comparable at each place; the product formula then gives two terms of comparable size at every place of S, and the paper combines them, a_ju_j + a_ku_k = au, with a new coefficient a of small height ("matching", pp. 1--2 and 6), reducing to four terms. Theorem 6 (p. 7) records the stronger form the proof gives, with coefficients allowed to vary subject to H(a_i) <= kappa_1 h(u)^kappa_2. Section 3 (pp. 7--9) applies matching to systems of S-unit equations (Theorem 7, p. 8).

The rest of the paper (Sections 4--7, pp. 10--29) proves Theorem 3. Let omega(N) count the nonnegative integer tuples (a,b,c,d) with N = 2^a 3^b + 2^c + 3^d, two tuples counting as one when their summand sets {2^a 3^b, 2^c, 3^d} agree (p. 2). The paper states D. J. Newman's question (Erdos and Graham, p. 80) as whether omega(N) is absolutely bounded, notes that Evertse, Gyory, Stewart and Tijdeman settled it affirmatively, and quotes Tijdeman and Wang's refinement as Theorem 2: omega(N) <= 4 for all N > N_0, for a constant N_0 that their proof does not make explicit (p. 2). Theorem 3 (p. 2) makes this explicit: omega(N) <= 4 for N >= 131082, <= 5 for N >= 19700, <= 6 for N >= 2316, <= 7 for N >= 786, <= 8 for N >= 300, and <= 9 for all N >= 1; omega(N) = 9 exactly for N in {41, 83, 89, 113, 137, 161, 227, 299}; the largest N with omega(N) = 5, 6, 7, 8 are 131081, 19699, 2315, 785; and the identities (4) for N = 2^a + 3^b give infinitely many N with omega(N) = 4. The proof finds the 1431 primitive five-term vanishing sums of terms +-2^alpha 3^beta with no vanishing subsums (Theorem 8, p. 15), places every N with a pair of representations having a vanishing subsum in three explicit special families, with bounds and extremal values of omega on each (Theorem 10, p. 20), and shows that outside those families omega(N) >= 3 holds only for ten listed N, each with omega(N) = 3 (Theorem 11, p. 23), the last step using matching and Theorem 6. Section 8 (p. 29) remarks that the arguments extend to rational N with exponents allowed to be negative.

Source: https://arxiv.org/abs/2308.05162.

Read status. Claims checked: Theorems 1, 3, 6, 8, 10 and 11 were read clause by clause on the printed pages (pp. 2, 7, 15, 19--20 and 23). Their proofs were read for structure only. The paper writes out in full one of the eighteen families behind Theorem 8 (pp. 16--18), one case of Theorem 10 (pp. 21--23) and the computationally hardest case of Theorem 11 (pp. 25--29), and says the others proceed similarly; none of the computations was checked, and nothing here is independently reviewed. As printed, Theorem 10's type I clause gives omega(N) = 4 for N = 2^a + 3^b with min{a,b} >= 2 with no lower bound on N, which the same clause's value omega(137) = 9 contradicts for small N (see its result page).

Bears on.

• #407: the paper presents Theorem 3 as an explicit answer to Newman's question as it states it, whether omega(N) is absolutely bounded: omega(N) <= 9 for every positive N, with the thresholds above. The problem page's w(n) counts quadruples (a,b,c,d), while the paper's omega(N) counts distinct summand sets. Theorems 1, 6, 8, 10 and 11 enter only as steps toward Theorem 3.

Results.

• Theorem 1 (p. 2): effective height bound for the nondegenerate solutions of five-term S-unit equations with |S| <= 3.
• Theorem 3 (p. 2): explicit bounds for omega(N), at most 9 for every N and at most 4 from 131082 on.
• Theorem 6 (p. 7): Theorem 1 with coefficients of height at most kappa_1 h(u)^kappa_2.
• Theorem 8 (p. 15): the 1431 primitive five-term vanishing sums of terms +-2^alpha 3^beta with no vanishing subsums.
• Theorem 10 (p. 20): pairs of representations with a vanishing subsum occur only for N in three special families, with bounds and extremal values of omega on each.
• Theorem 11 (p. 23): outside the special families, omega(N) >= 3 only for ten listed N, each with omega(N) = 3.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.