1 results
A new lower bound in the
$abc$ conjecture
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 2 / June 2024
- Published online by Cambridge University Press:
- 09 October 2023, pp. 369-378
- Print publication:
- June 2024
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- Article
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We prove that there exist infinitely many coprime numbers a, b, c with
$a+b=c$ and 
$c>\operatorname {\mathrm {rad}}(abc)\exp (6.563\sqrt {\log c}/\log \log c)$. These are the most extremal examples currently known in the 
$abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. Our work builds on that of van Frankenhuysen (J. Number Theory 82(2000), 91–95) who proved the existence of examples satisfying the above bound with the constant 
$6.068$ in place of 
$6.563$. We show that the constant 
$6.563$ may be replaced by 
$4\sqrt {2\delta /e}$ where 
$\delta $ is a constant such that all unimodular lattices of sufficiently large dimension n contain a nonzero vector with 
$\ell _1$-norm at most 
$n/\delta $.
