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Sep 29, 2016 at 16:36 history edited Noam D. Elkies CC BY-SA 3.0
Change X_M to X_B now that the OP made the corresponding (logical) change in the question.
Sep 29, 2016 at 2:21 vote accept Jernej
Sep 29, 2016 at 0:35 comment added Noam D. Elkies Likewise for $k=11$, $k=19$, and presumably an exercise to prove it for all larger prime powers congruent to $3 \bmod 4$ (though not $3$ itself).
Sep 28, 2016 at 23:39 history answered Noam D. Elkies CC BY-SA 3.0