Claims

../

1959_01_01_sekanina: Sekanina (Czechoslovak Math. J. 1959) proves that the squares have no tiling complement in the integers, so f(X) = X^2 does not answer the existence question; refereed.

2026_06_28_pipeline_math: The pipeline-math manuscript constructs a set A such that every integer is uniquely a member of A plus an integer thirteenth power, answering the existence question with f(X) = X^13.

2026_07_27_price: Price constructs, for every d at least 5, a computable set A_d such that every integer is uniquely a member of A_d plus a nonnegative d-th power; d = 6 answers the all-integer question.

2026_09_25_shan_xu_liang_dai_chen: Shan, Xu, Liang, Dai and Chen claim a set A such that every integer is uniquely a member of A plus an integer cube, and that no integer-valued quadratic polynomial admits such a complement.

---