We are looking the lower bound of the matrix B^n in the factorization of the Hilbert matrix of order n of the form H^n=C^nB^n, where C^n is the Cesaro matrix of order n.
Have a look at Article Lower Bounds on Matrix Factorization Ranks via Noncommutativ...
https://ccse.lbl.gov/Publications/sepp/matrixLowerBound/LBNL-50635.pdf
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