Every real number , has an essentially unique expansion as a Pierce series : where the form a strictly increasing sequence of positive integers. The expansion terminates if and only if is rational. Similarly, every positive real number has a unique expansion as an Engel series : where the form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates if and only if is rational. In this paper we obtain some new upper and lower bounds on the lengths of these series on rational inputs . In the case of the Engel series, this answers an open question of Erdös, Rényi, and Szüsz. However, our upper and lower bounds are widely separated.
@article{JTNB_1991__3_1_43_0, author = {P. Erd\"os and J. O. Shallit}, title = {New bounds on the length of finite pierce and {Engel} series}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {43--53}, publisher = {Universit\'e Bordeaux I}, volume = {3}, number = {1}, year = {1991}, doi = {10.5802/jtnb.41}, zbl = {0727.11003}, mrnumber = {1116100}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.41/} }
TY - JOUR TI - New bounds on the length of finite pierce and Engel series JO - Journal de Théorie des Nombres de Bordeaux PY - 1991 DA - 1991/// SP - 43 EP - 53 VL - 3 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.41/ UR - https://zbmath.org/?q=an%3A0727.11003 UR - https://www.ams.org/mathscinet-getitem?mr=1116100 UR - https://doi.org/10.5802/jtnb.41 DO - 10.5802/jtnb.41 LA - en ID - JTNB_1991__3_1_43_0 ER -
P. Erdös; J. O. Shallit. New bounds on the length of finite pierce and Engel series. Journal de Théorie des Nombres de Bordeaux, Volume 3 (1991) no. 1, pp. 43-53. doi : 10.5802/jtnb.41. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.41/
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