Let F_1(q),F_2(q),F_3(q),F_4(q) denote the four weighted truncated pentagonal-number series appearing in Merca's conjectures. The conjectures assert that [q^n]F_i(q)≥ 0 for every n≥ 0 and each i∈1,2,3,4. The weights include ∏_m≥ 1(1-q^2N_m), where N_m=m(1+nu_2(m)/2), and ∏_r≥ 1(q^2^r r;q^2^r+1r)_∞. The supplied source does not state the four truncations explicitly.

Merca’s truncated pentagonal-number-series positivity conjectures

Let $F_1(q),F_2(q),F_3(q),F_4(q)$ denote the four weighted truncated pentagonal-number series appearing in Merca's conjectures. The conjectures assert that $[q^n]F_i(q)\ge 0$ for every $n\ge 0$ and each $i\in\{1,2,3,4\}$ . The weights include $\prod_{m\ge 1}(1-q^{2N_m})$ , where $N_m=m\left(1+\nu_2(m)/2\right)$ , and $\prod_{r\ge 1}(q^{2^r r};q^{2^{r+1}r})_\infty$ . The supplied source does not state the four truncations explicitly.

References

Primary source

arXiv

Additional references

On the positivity of truncated pentagonal number series and some conjectures of Merca — arXiv — Manjil P. Saikia, Abhishek Sarma

partition-theory q-series

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

An unrefereed preprint claims to prove all four positivity conjectures, but independent verification has not been found.

Merca’s conjectures assert coefficient nonnegativity for four weighted truncations of pentagonal-number $q$ -series. The latest claim is by Manjil P. Saikia and Abhishek Sarma.

Known results

Andrews and Merca handled truncations after an odd number of terms; Kolitsch handled another truncation.
R. Mao and Y. reported an independent positivity result in $2015$ ; C. Wang and Y. reproved it in $2019$ .
A preprint at version $v3$ claims proofs of two Merca conjectures, Conjectures $13$ and $15$ .
Wright’s Circle Method establishes the conjectures for sufficiently large coefficient index, giving asymptotic rather than complete positivity.

September 2026 claimed proof

Saikia and Sarma claim nonnegativity for all four conjectured weighted truncations and introduce an orbit-telescoping method complementary to the Pólya–Szegő criterion. The result is an unrefereed preprint, with no reported counterexample or independent verification.

Current status (as of September 2026): The four conjectures are claimed proved in an unrefereed preprint, while the claim remains unverified.

Sources

No solutions have been posted yet.