The set of N-smooth integers is defined as the integers whose largest
prime factor is =N. I conjecture (and I'm sure I'm not the first to
do so) that for any N and k, the N-smooth integers have only finitely
many "gaps" of size =k.

(I suppose you could call a sequence with this property an "anti-
Cauchy sequence", since it is defined exactly like a Cauchy sequence
with the inequality reversed, so my conjecture can be stated "for all
N, the N-smooth integers are anti-Cauchy".)

However, that's not my question. My question is, has anyone ever come
up with a plausible heuristic bound for how high up, in terms of N and
k, you have to go for the N-smooth integers to always be more than k
apart from then on.

If that's too hard, I'd like a heuristic bound for how high you have
to go to get past all adjacent products of small primes. For example,

123200=64*25*7*11 = 2^6 * 5^2 * 7 * 11;
123201=729*169 = 3^6 * 13^2

Is this the largest example for N=13 and k=1?

-- Joe Shipman

On Jul 5, 9:30Êam, " wrote:
The set of N-smooth integers is defined as the integers whose largest
prime factor is =N. I conjecture (and I'm sure I'm not the first to
do so) that for any N and k, the N-smooth integers have only finitely
many "gaps" of size =k.

(I suppose you could call a sequence with this property an "anti-
Cauchy sequence", since it is defined exactly like a Cauchy sequence
with the inequality reversed, so my conjecture can be stated "for all
N, the N-smooth integers are anti-Cauchy".)

However, that's not my question. My question is, has anyone ever come
up with a plausible heuristic bound for how high up, in terms of N and
k, you have to go for the N-smooth integers to always be more than k
apart from then on.

If that's too hard, I'd like a heuristic bound for how high you have
to go to get past all adjacent products of small primes. For example,

123200=64*25*7*11 = 2^6 * 5^2 * 7 * 11;
123201=729*169 = 3^6 * 13^2

Is this the largest example for N=13 and k=1?

-- Joe Shipman

Replying to myself:

The Thue-Siegel-Roth theorem appears to imply my conjecture that the N-
smooth integers are anti-Cauchy, though nonconstructively; the simpler
conjecture that there are only finitely many consecutive pairs with
all prime factors =N was proven, constructively, by Stoermer. (Or,
if you have Norwegian fonts in your browser, St¿rmer.)

See he

http://www.research.att.com/~njas/sequences/A117581

and he

http://en.wikipedia.org/wiki/Stormer%27s_theorem


Neither source gives a tight heuristic bound on the size of the
largest pair for each N, although bounds for the number of pairs are
discussed.