Template:Sequence of the Day for July 21

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 2 m)

. Even-indexed powers of two (A000302) are squares of a smaller power of two, and therefore

2√  2 m  +  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.