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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Collatz_ABC.pdf (531.88 Ko)
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LBH_misses_j30000_all.pdf (177.3 Ko)
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mu_abc_hits_all.pdf (60.14 Ko)
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