Abstract
In this article, we prove that the density of integers a, b such that \(a^4+b^3\) is squarefree, when ordered by \(\max \{|a|^{1/3},|b|^{1/4}\}\), equals the conjectured product of the local densities. We show that the same is true for polynomials of the form \(\beta a^4 + \alpha b^3\) for any fixed integers \(\alpha \) and \(\beta \). We give an exact count for the number of pairs (a, b) of integers with \(\max \{|a|^{1/3},|b|^{1/4}\}<X\) such that \(\beta a^4 + \alpha b^3\) is squarefree, with a power-saving error term.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and material
Not applicable.
References
Apostal, T.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York (1976)
Ash, A., Brakenhoff, J., Zarrabi, T.: Equality of polynomial and field discriminants. Exp. Math. 16, 367–374 (2007)
Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums. Wiley, New York (1998)
Bhargava, M., Shankar, A.: Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Ann. Math. (2) 181, 191–242 (2015)
Bhargava, M., Shankar, A., Wang, X.: Squarefree values of polynomial discriminants I. Invent. Math. (2022). https://doi.org/10.1007/s00222-022-01098-w
Bhargava, M., Shankar, A., Wang, X.: Squarefree values of polynomial discriminants II. Preprint
Granville, A.: ABC allows us to count squarefrees. Int. Math. Res. Not. 19, 991–1009 (1998)
Greaves, G.: Power-free values of binary forms. Q. J. Math. Oxford Ser. (2) 43, 45–65 (1992)
Shankar, A., Wang, X.: Rational points on hyperelliptic curves having a marked non-Weierstrass point. Compos. Math. 154(1), 188–222 (2018)
Hooley, C.: On the square-free values of cubic polynomials. J. Reine Angew. Math. 229, 147–154 (1968)
Hooley, C.: On the power-free values of polynomials in two variables: II. J. Number Theory 129(6), 1443–1455 (2009)
Kowalski, J.: On the proportion of squarefree numbers among sums of cubic polynomials. Ramanujan J. 54(2), 343–354 (2021)
Murty, R., Paston, H.: Counting squarefree values of polynomials with error term. Int. J. Number Theory 10(7), 1743–1760 (2014)
Poonen, B.: Squarefree values of multivariable polynomials. Duke Math. J. 118(2), 353–373 (2003)
Heath-Brown, D.: Power-free values of polynomials. Q. J. Math. 64, 177–188 (2013)
Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications 53, Providence, RI (2004)
Shankar, A., Tsimerman, J.: Counting $S_5$-fields with a power saving error term. Forum Math. Sigma 2, e13 (2014)
Acknowledgements
It is a pleasure to thank Manjul Bhargava for many helpful comments. The first named author is supported by the University of Waterloo through an MURA project. The second named author is supported by an NSERC Discovery Grant.
Funding
The first named author is supported by the University of Waterloo through an MURA project. The second named author is supported by an NSERC Discovery Grant.
Author information
Authors and Affiliations
Contributions
Not applicable.
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Code availability
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sanjaya, G.C., Wang, X. On the squarefree values of \(a^4+b^3\). Math. Ann. 386, 1237–1265 (2023). https://doi.org/10.1007/s00208-022-02404-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-022-02404-w