The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits

Jon Grantham; Hester Graves

The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits

Number Theory 2021-05-27 v4

Authors: Jon Grantham , Hester Graves

Abstract

Assuming the abc conjecture with $\epsilon=1/6$ , we use elementary methods to show that only finitely many $s$ -Cullen numbers are repunits, aside from two known infinite families. More precisely, only finitely many positive integers $s$ , $n$ , $b$ , and $q$ with $s,b \geq 2$ and $n,q \geq 3$ satisfy $C_{s,n} = ns^n + 1 = \frac{b^q -1}{b-1}.$

Keywords

abc conjecture s-cullen numbers repunits diophantine equations elementary methods

Cite

@article{arxiv.2009.04052,
  title  = {The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits},
  author = {Jon Grantham and Hester Graves},
  journal= {arXiv preprint arXiv:2009.04052},
  year   = {2021}
}

Comments

v4: fixed typo, added journal reference v3: Significantly improved the main result v2: Improved formatting, better Masser reference

The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits

Abstract

Keywords

Cite

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