Conjecture on the Hilbert symbol and the coefficients Uk(u)U_k^{(u)}Uk(u)
The conjecture. For 0≤k≤⌈j−i02⌉0\leq k\leq \left\lceil\frac{j-i_0}{2}\right\rceil0≤k≤⌈2j−i0⌉, the coefficients satisfy the stated congruences for Uk(u)U_k^{(u)}Uk(u) and Uk′U_k'Uk′, and consequently
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