Article Dans Une Revue Experimental Mathematics Année : 2025

Are the Collatz and abc conjectures related?

Résumé

The Collatz and $abc$ conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the $abc$ conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the $abc$-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset.

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hal-04142385 , version 1 (26-06-2023)
hal-04142385 , version 2 (03-07-2023)
hal-04142385 , version 3 (21-10-2024)
hal-04142385 , version 4 (20-10-2025)

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Olivier Rozier. Are the Collatz and abc conjectures related?. Experimental Mathematics, 2025, pp.1-10. ⟨10.1080/10586458.2025.2470938⟩. ⟨hal-04142385v3⟩
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