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Updated Oct 8

Problem 1081

claims/: The 3 claim pages of Problem 1081, one per claimant's result; the problem's standing derives from them.

Statement. Let $A(x)$ count the number of $n\leq x$ which are the sum of two squarefull numbers (a number $m$ is squarefull if $p\mid m$ implies $p^2\mid m$). Is it true that

A(x) \sim c \frac{x}{\sqrt{\log x}}

for some $c>0$ ?

Status. Disproved.

Source. erdosproblems.com/1081, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1081, https://www.erdosproblems.com/1081.

References.

[BaBr94] Baker, R. C. and Brüdern, J., On sums of two squarefull numbers. Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 1, 1-5.
[Bl04] Blomer, Valentin, Binary quadratic forms with large discriminants and sums of two squareful numbers. J. Reine Angew. Math. 569 (2004), 213-234.
[BlGr06] Blomer, Valentin and Granville, Andrew, Estimates for representation numbers of quadratic forms. Duke Math. J. 135 (2006), no. 2, 261-302.
[Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
[Od81] Odoni, R. W. K., A problem of Erdős on sums of two squarefull numbers. Acta Arith. 39 (1981), no. 2, 145-162.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (11)

Every large integer is a sum of at most three powerful numbers Odoni disproves the square-root-of-log asymptotic Blomer's lower bound refutes the square-root-of-log asymptotic Blomer and Granville pin the order of sums of two powerful numbers problems/diophantine_problems/E1081/claims Diophantine Problems and Powers diophantine_problems/blomer_2006_estimates_representation_numbers_quadratic_forms Corollary 2 (p. 8): sums of two powerful numbers up to x number x/(log x)^(1-2^(-1/3)) up to powers of log log x Theorem 5 (p. 7): the β-th moment of r_f(n) is x (log x)^(E(κ,β)+o(1)) for D ≤ (log x)^L Theorem 6 (p. 8): upper bounds for N_f(x) in three ranges of κ, and the lower bound for κ ≤ 1/2 - ε diophantine_problems/odoni_1981_problem_erdos_sums_two_squarefull_numbers

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