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A positive integer is said to be a rebel number if it is the product of numbers none of which share any of the original number´s digits. Thus 10 = 2 x 5 is a rebel number, while none of the primes is. Ten consecutive integers cannot be all rebel numbers because one of them will be an odd multiple of 5 (thus terminating in 5) and with at least one factor terminating in 5. Five consecutive rebel numbers are 666 to 670.

Do nine consecutive positive integers all of which are rebel exist? Do they exist infinitely often?

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migrated from mathoverflow.net Apr 27 '15 at 15:43

This question came from our site for professional mathematicians.

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    $\begingroup$ Why do you care about the digits base 10? Usually, this indicates that the problem is recreational mathematics and unlikely to be connected to anything rather than what research mathematicians study. $\endgroup$ – Douglas Zare Apr 27 '15 at 14:05
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    $\begingroup$ Recreational of not, elementary or advanced, pure or applied, mathematics is one. I care about it all! $\endgroup$ – Bernardo Recamán Santos Apr 27 '15 at 14:25
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    $\begingroup$ I enjoy your contributions to this forum..However, it could be argued that math.stackexchange is a better forum for some of them. We will see how the community treats this contribution. $\endgroup$ – The Masked Avenger Apr 27 '15 at 14:30
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    $\begingroup$ I usually applaud when other people have broad interests, but I don't think this qualifies as on-topic. I don't think it counts as "the sorts of questions you come across when you're writing or reading articles or graduate-level books." Does anyone else study "rebel numbers?" Quick Google searches turned up pages on civil wars. Your definition seems unmotivated and unnatural to me, like studying not the largest eigenvalue but the one that is alphabetically first in Tagalog. What fields of mathematics other than elementary number theory do you expect would be helpful for studying these numbers? $\endgroup$ – Douglas Zare Apr 27 '15 at 14:57
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Up to $10^5$, the starts of runs of $5$ consecutive rebel numbers are $666, 686, 1908, 2208, 6666, 7886, 11106, 16166, 21998, 66668, 66786, 70886, 77006$, and there are no runs of more than $5$. This is not an answer, but some evidence pointing to the possiblity that runs of more than $5$ are very rare and maybe nonexistent. The sequence $666, 686, 1908, \ldots$ does not seem to be in the OEIS. Maybe it should be.

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    $\begingroup$ Freddy Barrera of Colombia Aprendiendo has found a run of 7:6777666, 6777667, 6777668, 6777669, 6777670, 6777671, 6777672. $\endgroup$ – Bernardo Recamán Santos Apr 28 '15 at 12:49
  • $\begingroup$ Barrera also found two other runs of 6 consecutive rebel numbers: [11666666, 11666667, 11666668, 11666669, 11666670, 11666671] and [16161609, 16161610, 16161611, 16161612, 16161613, 16161614]. $\endgroup$ – Bernardo Recamán Santos Apr 28 '15 at 14:07

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