Odoni 1981 problem erdos sums two squarefull numbers

../

---

Odoni, R. W. K., A problem of {E}rdős on sums of two squarefull numbers. Acta Arith. 39 (1981), no. 2, 145--162; DOI 10.4064/aa-39-2-145-162.

Erdos asked whether the count of integers up to x that are sums of two squarefull (powerful) numbers is asymptotically C x (log x)^{-1/2}, the quantity suggested by analogy with Landau's theorem for sums of two squares. Odoni answers in the negative with Theorem 1: there are positive constants a, beta, gamma such that card(U ∩ [1,x]) > a x (log x)^{-1/2} exp(beta log log x / log log log x) for all x > gamma, where U is the set of sums of two squarefull numbers. The method recasts the problem as representation by the family of binary quadratic forms F_{mn}(x,y) = m^3 x^2 + n^3 y^2 with (m,n)=1, then restricts to the squarefree members represented properly by forms F_{pq} of discriminant -4p^3q^3 with p_0 < p < q <= y(x), where the genus theory is tractable; a combinatorial inclusion-exclusion inequality (0.2) reduces matters to a lower bound for card(F_{pq} ∩ [1,x]) (Theorem 2, giving c x (log x)^{-1/2} uniformly for suitable D) and an upper bound for the pairwise overlaps (Theorem 3), combined in section 14. Sections 1 onward assemble the classical theory of primitive binary forms, genera and rational equivalence used throughout. This settles the asymptotic question raised in problem 1081 by showing the naive count is too small.

Source: https://eudml.org/doc/205760. The image-only scan shows no copyright or license line on its first and last pages; IMPAN's volume listing offers the article's PDF "Free download under CC-BY license", no version named (https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/39, read 2026-10-02; the article's own page was not opened), so the term is the Creative Commons Attribution license without a version; the site footer "Copyright © 2026 by IMPAN. All rights reserved." is the website's, not the article's.

Bears on. #1081

Results to transcribe.

• Theorem 1: For the set U of sums of two squarefull numbers there are a, beta, gamma > 0 with card(U ∩ [1,x]) > a x (log x)^{-1/2} exp(beta log log x / log log log x) for x > gamma.
• Theorem 2: If D = 4p^3q^3 with q > p > p_0 and (log D)^{c_53 log D} <= log x, where x > x_0, then card(F_{pq} ∩ [1,x]) > c_57 x (log x)^{-1/2}, the c_n being positive absolute constants.
• Theorem 3: For sufficiently large x, if (log Delta)^{c_61 log Delta} <= log x, where Delta = 16 p^3 q^3 p'^3 q'^3 with p, q, p', q' > p_0, then card(F_{pq} ∩ F_{p'q'} ∩ [1,x]) <= c_59 (log Delta)^{c_60} x (log x)^{-3/4}, bounding overlaps between forms.
• Lemma 1.1 (p. 147): "The positive integer n (prime to 2pq) is properly represented by some primitive form of discriminant -4p^3q^3 if and only if (-pq) is a square (mod n)." Here p and q are primes with p_0 < p < q <= y.