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Problems & Puzzles:
Puzzles
Puzzle 1257 2^p-7,
all composites if p is prime?
On
Feb 4, 2026,
Sebastián Martín Ruiz sent this interesting puzzle.
A ccording
to OEIS
A059609*,
2^n-7 is prime for the following
twelve
known n
values:
39, 715, 1983, 2319, 2499, 3775, 12819,
63583, 121555, 121839, 468523 & 908739
**, . You know what?
all of these twelve n values, are
composite integers!
Q1. Can
you find the next n value?
According to OEIS
A059609,
it´s valid suppose that 2^p-7 is composite for at least all
p,
prime numbers <= 908731 (the largest
prime less than 908739).
Q2. Then, prove
that for p=prime>2, 2^p-7 is composite or
find a counterexample.
_________
*
A059609
has a long story, covering
22 years:
Author: Andrey V. Kulsha,
Feb 02
2001. EXTENSIONS: a(8) from Henri
Lifchitz; a(9)-a(10) from Gary Barnes, added
by Max Alekseyev, Feb 09 2012; a(11) from
Lelio R Paula, added by Max Alekseyev, Oct
25 2015; a(12) from Jon Grantham,
Aug 09 2023.
** The integer 2^908739-7 has around 273558
digits in decimal numerical system.
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