MathDB

Continued fraction conjecture for −π/4-\pi/4

A continued fraction has partial numerators ana_n and partial denominators bnb_n, with the displayed initial terms determining the pattern. Continued fraction conjecture for −π/4-\pi/4. The constant −π/4-\pi/4 admits the continued fraction representation

−π4=1−1+12−3+22−5+32−7+…-\frac{\pi}{4}=\cfrac{1}{-1+\cfrac{1^2}{-3+\cfrac{2^2}{-5+\cfrac{3^2}{-7+\dots}}}}

where

an=(n−1)2,bn=−(2n−1),n∈N.a_n=(n-1)^2,\qquad b_n=-(2n-1),\qquad n\in\mathbb{N}.

This is presented as a non-canonical continued-fraction identity in the context of symbolic induction and the search for alternative representations of transcendental constants; the supplied material gives no evidence that it has been proved or disproved.

References

Primary source

Chao Wang, “A Family of Continued Fraction Identities for Arctangent Values”, arXiv:2601.11892 (2026).

arctangentcontinued-fractionstranscendental-numbers
Posted · edited

Progress summary

Refreshed
Claimed solved

A January 2026 paper claims to prove the continued-fraction identity, but the proof has not been independently verified.

The problem asks whether the displayed continued fraction really equals −π/4-\pi/4. The identity was presented as a conjecture associated with the Ramanujan Machine.

January 2026 claimed proof

A paper titled A Proof of the Continued Fraction Identity −π/4=⋯-\pi/4=\cdots claims a rigorous, self-contained proof. It identifies the fraction with the Gauss continued fraction for arctan⁡(−1)\arctan(-1) and uses a constant equivalence transformation to obtain the stated coefficients, concluding convergence to −π/4-\pi/4. No counterexample, withdrawal, or independent verification is reported.

Current status (as of September 2026): A complete proof is claimed in the January 2026 arXiv paper, but the identity remains independently unverified.

Sources

Solutions 0

No solutions have been posted yet.