A Strange Sequence not in the OEoIS
Don Reble
djr at nk.ca
Wed Mar 26 20:27:38 CET 2008
> One can search for numbers x such that P(x) (P a polynomial)
> is B-smooth (for moderate B) extremely fast (e.g. P=x^2+-1
> and using the first 64 primes: 500 Million tests per second).
>
> However, determining the _largest_ number x such that is, say,
> B-smooth with P(x):=x^2+-1 is not possible:
> I'd like to see a proof that the set is even _finite_!
The Lehmer reference of A002072 may be helpful.
The ideas are, that adjacent smooth numbers correspond to Pell
equation solutions; that Pell equation solutions are found amidst
Lucas sequences; and that Lucas sequence terms have predictable
divisibilities.
Generally, one can predict which terms of a Lucas sequence are
divisible by any prime power, without computing the terms. And one
can compute an upper bound on the largest smooth factor of a Lucas
term, as a function of the term's position. But a Lucas sequence
grows faster than such a bound, so all terms beyond some point are
non-smooth.
--
Don Reble djr at nk.ca
--
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