Publications

1. Bajpai, P., & Bennett, M. (2024). Effective S-unit Equations Beyond 3 Terms : Newman’s Conjecture. Acta Arithmetica, To appear.
   BIB ABSTRACT

   @article{BajBen24,
     author = {Bajpai, Prajeet and Bennett, Mike},
     journal = {Acta Arithmetica},
     number = {},
     pages = {},
     title = {Effective $S$-unit Equations Beyond 3 Terms : Newman's Conjecture},
     volume = {To appear.},
     year = {2024}
   }
   

   We show how to effectively solve 5-term S-unit equations when the set of primes S has cardinality at most 3, and use this to provide an explicit answer to an old question of D.J. Newman on representations of integers as sums of S-units.

2. Bajpai, P., & Bugeaud, Y. (2024). Effective Approximation to Complex Algebraic Numbers by Algebraic Numbers of Bounded Degree. Transactions of the American Mathematical Society, 377, 5247–5269.
   BIB ABSTRACT

   @article{BajBu24,
     author = {Bajpai, Prajeet and Bugeaud, Yann},
     journal = {Transactions of the American Mathematical Society},
     pages = {5247-5269},
     title = {Effective Approximation to Complex Algebraic Numbers by Algebraic Numbers of Bounded Degree},
     volume = {377},
     year = {2024}
   }
   

   We establish the first effective improvements on the Liouville inequality for approximation to complex non-real algebraic numbers by complex algebraic numbers of degree at most 4.

3. Bajpai, P., Bennett, M. A., & Chan, T. H. (2024). Arithmetic progressions in squarefull numbers. International Journal of Number Theory, 20(01), 19–45.
   BIB ABSTRACT

   @article{BajBenChan,
     author = {Bajpai, Prajeet and Bennett, Michael A. and Chan, Tsz Ho},
     title = {Arithmetic progressions in squarefull numbers},
     journal = {International Journal of Number Theory},
     volume = {20},
     number = {01},
     pages = {19-45},
     year = {2024}
   }
   

   We answer a number of questions of Erdős on the existence of arithmetic progressions in k-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the k-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the abc-conjecture of Masser and Oesterlé.

4. Bajpai, P. (2023). Effective methods for norm-form equations. Mathematische Annalen, 387(3), 1271–1288.
   BIB ABSTRACT

   @article{Baj23,
     author = {Bajpai, Prajeet},
     journal = {Mathematische Annalen},
     number = {3},
     pages = {1271--1288},
     title = {Effective methods for norm-form equations},
     volume = {387},
     year = {2023}
   }
   

   While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to address the case of three variables over totally complex and Galois number fields. In this paper, we extend his results to effectively resolve several new classes of norm-form equations. In particular, we completely and effectively settle the question of norm-form equations over totally complex Galois sextic fields.

5. Bajpai, P., & Bennett, M. (2022). A note on pencils of norm-form equations. Acta Arithmetica, 203, 19–26.
   BIB ABSTRACT

   @article{BajBenPencil,
     author = {Bajpai, Prajeet and Bennett, Mike},
     journal = {Acta Arithmetica},
     pages = {19-26},
     title = {A note on pencils of norm-form equations},
     volume = {203},
     year = {2022}
   }
   

   We find all solutions to a parametrized family of norm-form equations x^3−(t^3−1)y^3+3(t^3−1)xy+(t^3−1)^2=±1 studied by Amoroso, Masser and Zannier. Our proof relies upon an appeal to lower bounds for linear forms in logarithms and various elementary arguments.