JournalsrmiVol. 37, No. 2pp. 723–748

On the representation of kk-free integers by binary forms

  • Cameron L. Stewart

    University of Waterloo, Canada
  • Stanley Yao Xiao

    University of Toronto, Canada
On the representation of $k$-free integers by binary forms cover
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Abstract

Let FF be a binary form with integer coefficients, non-zero discriminant and degree dd with dd at least 33 and let rr denote the largest degree of an irreducible factor of FF over the rationals. Let kk be an integer with k2k \geq 2 and suppose that there is no prime pp such that pkp^k divides F(a,b)F(a,b) for all pairs of integers (a,b)(a,b). Let RF,k(Z)R_{F,k}(Z) denote the number of kk-free integers of absolute value at most ZZ which are represented by FF. We prove that there is a positive number CF,kC_{F,k} such that RF,k(Z)R_{F,k}(Z) is asymptotic to CF,kZ2/dC_{F,k} Z^{2/d} provided that kk exceeds 7r/18{7r}/{18} or (k,r)(k,r) is (2,6)(2,6) or (3,8)(3,8).

Cite this article

Cameron L. Stewart, Stanley Yao Xiao, On the representation of kk-free integers by binary forms. Rev. Mat. Iberoam. 37 (2021), no. 2, pp. 723–748

DOI 10.4171/RMI/1213