Wang–Sun conjecture on quadratic representations of practical numbers

A positive integer $m$ is called practical if every positive integer $r\le m$ can be expressed as a sum of distinct positive divisors of $m$ . The Wang–Sun conjecture asserts that, for all positive integers $a,b,c$ satisfying $2\nmid a$ , $2\nmid b$ , and $2\mid c$ , there are infinitely many positive integers $n$ such that $an^2+bn+c$ is practical. It further asserts that, when $a=1$ , there exists an integer $n$ with $1<n\le\max\{b,c\}$ such that $n^2+bn+c$ is practical.

References

Primary source

arXiv

Additional references

A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to complete the conjecture, but independent verification has not yet appeared.

Wang and Sun conjectured in 2018 that quadratic polynomials with positive integer coefficients produce practical numbers infinitely often under specified parity conditions. The conjecture also includes a stronger bounded-value assertion when the leading coefficient is $1$ .

Known results

The 2018 paper proved that, once one value $n>1$ works in the leading-coefficient- $1$ case, infinitely many such values follow.
A December 2022 paper gave a necessary-and-sufficient criterion for infinitely many practical values of a quadratic polynomial.
The same paper proved the infinite-value assertion when $a$ and $b$ are odd and $c$ is even.

August 2026 claimed completion

An August 2026 preprint, A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers, claims to prove the remaining case for odd $b$ and even $c$ , thereby completing the conjecture together with earlier results. This completion claim is unverified.

Current status (as of August 2026): The conjecture is claimed solved by an unrefereed preprint, but the claimed completion has not been independently verified.

Sources

No solutions have been posted yet.