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Template:Sequence of the Day for July 21

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Intended for: July 21, 2012

Timetable

  • First draft entered by Alonso del Arte on April 29, 2011 based on a verbatim copy of a write-up from November 13, 2010. ✓
  • Draft reviewed by Alonso del Arte on July 16, 2011
  • Draft approved by Daniel Forgues on July 20, 2011
Yesterday's SOTD * Tomorrow's SOTD

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A175607 Largest number
x
such that the greatest prime factor of
x 2  −  1
is
pn , n   ≥   1
.
{ 3, 17, 161, 8749, 19601, 246401, 672281, ... }
It surprised me, but after a little reflection, it makes a lot of sense: for any prime
p
, there is a short (that is to say, finite) list of numbers
x
such that
x 2  −  1
has
p
as its greatest prime factor. (See: Greatest prime factor of n 2  −  1, n   ≥   2.) For example, for
p1 = 2
, we find that 3 2  −  1 = 8 = 2 3. No larger number satisfies
k 2   ≡   1  (mod 2m)
. Even-indexed powers of two (A000302) are squares of a smaller power of two, and therefore
2  2m  +  1
is not an integer if
m
is even. The reason why higher odd-indexed powers of two (A004171) don’t work for this purpose is, as so many math textbooks say, left as an exercise for the Reader. According to Artur Jasinski, every prime
p
has a corresponding
x
with the property described here.