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Statement

Theorem 3 (p. 3). "For any set of natural numbers $A$, there are at least $|A|$ natural numbers in the set $\{ab/\gcd(a,b)^2 : a,b\in A\}$."

The paper offers it as the symmetric version of its Unsolved problem, since $a/\gcd(a,b)$ is not symmetric in $a$ and $b$, and calls it "a best possible result" (p. 3): taking $A=\{d:d\mid n\}$ gives equality, as the Remark on p. 3 notes. The statement places no finiteness hypothesis on $A$; its vector form, Theorem 4, is stated for finite sets.

In exponent vectors over the primes dividing members of $A$, the number $ab/\gcd(a,b)^2$ has exponent vector $d(\mathbf a,\mathbf b)=(|a_1-b_1|,\dots,|a_n-b_n|) =\delta(\mathbf a,\mathbf b)+\delta(\mathbf b,\mathbf a)$, and the paper states that Theorem 3 is equivalent to Theorem 4 (p. 3).

Equality. The Remark (p. 3) announces that equality holds only for the sets $A$ of integers $bq_1^{i_1}\cdots q_k^{i_k}$, where $q_j=r_j/s_j\ne1$ are positive rationals with $\gcd(r_j,s_j)=1$ and $\gcd(r_is_i,r_js_j)=1$ for $i\ne j$, the exponents satisfy $\ell_j\le i_j\le u_j$ for some bounds $\ell_j,u_j$, the paper writes $(i_1,\dots,i_k)\in S$ for a subgroup $S$ of $(\mathbb Z/2\mathbb Z)^k$ (the exponents taken modulo 2), and $b$ makes all these numbers integers. It gives $A=\{md^2:d\mid n,\ m=1\text{ or }b\}$, with squarefree $b>1$ dividing $n$, as a further example. The Remark defers the proof to section 3 (pp. 6--7), which states the characterization in vector form as Proposition 1 (p. 6) and only sketches its proof.

Source. A. Granville and F. Roesler, The set of differences of a given set, Amer. Math. Monthly 106 (1999), no. 4, 338--344; Theorem 3 and the Remark on p. 3 of the authors' eight-page preprint, read on the page image. The journal version was not compared.

Read depth. Claims checked: the statement, the Remark and the reduction to Theorem 4 were read clause by clause on the page image of p. 3. The equality characterization (Proposition 1, pp. 6--7) was read for its statement only.

Proof pointer

Equivalent to Theorem 4 through the exponent-vector translation above; the proof of Theorem 4 is in section 2 (pp. 4--5).

Dependencies

Theorem 4.

Bears on

None recorded. The quantity is the symmetric analogue of the ratio count of Problem 539; the paper derives no bound on that problem's $h(n)$ from it.