A Strange Sequence not in the OEoIS

[Don Reble]

> 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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