Report on Zhi-Wei Sun's 1-3-5 conjecture and some of its refinements.

We report here on the computational verification of Zhi-Wei Sun's "1-3-5 conjecture" for all natural numbers up to 105 103 560 126. This, together with a result of two of the authors, completes the proof of that conjecture. Furthermore, the computations made in the verification process of the 1-3-5 conjecture revealed a refinement, which we state as a separate conjecture at the end of the paper. [ABSTRACT FROM AUTHOR]