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SUMMARY:Thomas Bloom (University of Manchester)
DTSTART:20250611T150000Z
DTEND:20250611T160000Z
DTSTAMP:20260418T065818Z
UID:LNTS/166
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/166/">N
 umbers with small digits in multiple bases</a>\nby Thomas Bloom (Universit
 y of Manchester) as part of London number theory seminar\n\nLecture held i
 n K2.40\, King's Building\, King's College London\, WC2R 2LS.\n\nAbstract\
 nAn old conjecture of Graham asks whether there are infinitely many intege
 rs $n$ such that $\\binom{2n}{n}$ is coprime to 105. This is equivalent to
  asking whether there are infinitely many integers which only have the dig
 its 0\,1 in base 3\, 0\,1\,2 in base 5\, and 0\,1\,2\,3 in base 7. In gene
 ral\, one can ask whether there are infinitely many integers which only ha
 ve 'small' digits in multiple bases simultaneously. For two bases this was
  established in 1975 by Erdos\, Graham\, Ruzsa\, and Straus\, but the case
  of three or more bases is much more mysterious. I will discuss recent joi
 nt work with Ernie Croot\, in which we prove that (assuming the bases are
  sufficiently large) there are infinitely many integers such that almost a
 ll of the digits are small in all bases simultaneously.\n
LOCATION:https://researchseminars.org/talk/LNTS/166/
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