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Yuta is correct, Størmer's method, or preferably, D.H. Lehmer's 1963 refinement of that method, applies to a finite set of primes.

If the set contains $k$ primes, then you have $2^k-1$ Pell equations to solve (Størmer's original method involved solving $3^k$ Pell's.

I have had a particular interest in this subject for some years, and have provided most of the high-end values at OEIS, see http://oeis.org/A002072. I intend to raise some questions arising from this work in a new posting here at mathoverflow.

But to Yuta's original question, the only way I can think of to identify smooth numbers in a particular range is to use a smart sieving algorithm. But if the interval is really large, and the number of primes is also very large, this may not be practical.

By the way, in Lehmer's paper, the ø in Størmer "Størmer" appears as an "o" with an umlaut, Störmer", I have yet to determine whether Wiki which is mistaken or correct! I suspect Lehmer was !probably correct.

2 added 158 characters in body

Yuta is correct, Stormer's Størmer's method, or preferably, D.H. Lehmer's 1963 refinement of that method, applies to a finite set of primes.

If the set contains $k$ primes, then you have $2^k-1$ Pell equations to solve (Stormer's Størmer's original method involved solving $3^k$ Pell's.

I have had a particular interest in this subject for some years, and have provided most of the high-end values at OEIS, see http://oeis.org/A002072. I intend to raise some questions arising from this work in a new posting here at mathoverflow.

But to Yuta's original question, the only way I can think of to identify smooth numbers in a particular range is to use a smart sieving algorithm. But if the interval is really large, and the number of primes is also very large, this may not be practical.

By the way, in Lehmer's paper, the ø in Størmer appears as an "o" with an umlaut, I have yet to determine whether Wiki is mistaken or Lehmer was!

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Yuta is correct, Stormer's method, or preferably, Lehmer's 1963 refinement of that method, applies to a finite set of primes.

If the set contains $k$ primes, then you have $2^k-1$ Pell equations to solve (Stormer's original method involved solving $3^k$ Pell's.

I have had a particular interest in this subject for some years, and have provided most of the high-end values at OEIS, see http://oeis.org/A002072. I intend to raise some questions arising from this work in a new posting here at mathoverflow.

But to Yuta's original question, the only way I can think of to identify smooth numbers in a particular range is to use a smart sieving algorithm. But if the interval is really large, and the number of primes is also very large, this may not be practical.