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Updated Oct 8

Problem 407

claims/: The 3 claim pages of Problem 407, one per claimant's result; the problem's standing derives from them.

Statement. Let $w(n)$ count the number of solutions to

n=2^a+3^b+2^c3^d

with $a,b,c,d\geq 0$ integers. Is it true that $w(n)$ is bounded by some absolute constant?

Status. Proved: the site's label; its commentary credits Evertse, Győry, Stewart and Tijdeman with the proof. The frontmatter standing is derived from the accepted claim pages their proof, Tijdeman and Wang's bound of four and Bajpai and Bennett's effective bounds, each accepted on the site's credit and, for the two journal papers, on their refereed publication.

Source. erdosproblems.com/407, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #407, https://www.erdosproblems.com/407.

References.

[BaBe24] Bajpai, Prajeet and Bennett, Michael A., Effective $S$ -unit equations beyond three terms: Newman's conjecture. Acta Arith. (2024), 421-458.
[EGST88] Evertse, J.-H. and Győry, K. and Stewart, C. L. and Tijdeman, R., $S$ -unit equations and their applications. New advances in transcendence theory (Durham, 1986) (1988), 110-174.
[TiWa88] Tijdeman, R. and Wang, Lian Xiang, Sums of products of powers of given prime numbers. Pacific J. Math. (1988), 177-193.

Formalization. Statement in formal-conjectures, added 2026-09-20; its formal_proof attribute names a Lean development in an outside repository, recorded on the claim page of the proof it formalizes. Nothing was built or audited here.

Progress

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Known Results

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Linked library material

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Linked from (23)

Tijdeman and Wang's bound of four representations Evertse, Győry, Stewart and Tijdeman's proof of Newman's conjecture Bajpai and Bennett's effective bounds for Newman's problem problems/diophantine_problems/E0407/claims Diophantine Problems and Powers diophantine_problems/bajpai_2024_effective_unit_equations_beyond_three_terms Theorem 1 (p. 2): effective heights for five-term S-unit equations when |S| <= 3 Theorem 10 (p. 20): representations with a vanishing subsum come from three special families Theorem 11 (p. 23): outside the special families, three representations occur for ten N only Theorem 3 (p. 2): explicit bounds for the number of representations N = 2^a 3^b + 2^c + 3^d Theorem 6 (p. 7): Theorem 1 with coefficients of polynomially bounded height Theorem 8 (p. 15): the 1431 primitive five-term vanishing sums of terms ±2^α 3^β diophantine_problems/evertse_schlickewei_schmidt_2002_linear_equations_multiplicative_group Theorem 1.1 (p. 2): a bound for nondegenerate solutions of a linear equation in a group of finite rank Theorem 2.1 (p. 9): solutions near a group of finite rank lie in few subspaces diophantine_problems/tijdeman_1988_sums_products_powers_given_prime_numbers Lemma 4 (p. 186): the version of the Main Theorem on S-unit equations used for Theorem 4 Theorem 1 (pp. 178--179): the twelve non-trivial integer solutions of 2^x 3^y + 1 = 2^z + 3^w Theorem 2 (p. 179): the eight non-trivial integer solutions of 2^x 3^y + 2^z = 3^w + 1 Theorem 3 (p. 179): the nine non-trivial integer solutions of 2^x 3^y + 3^w = 2^z + 1 Theorem 4 (p. 186): beyond some M, a rational has at most four distinct representations 2^alpha 3^beta + 2^gamma + 3^delta Theorem 5 (p. 190): boundedly many representations as a sum of n products of powers of given primes Theorem 6 (p. 190): boundedly many representations w_1 s_1 + ... + w_n s_n by S-units of a number field

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