Difference between revisions of "2002 Indonesia MO Problems/Problem 6"
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] |
Latest revision as of 00:10, 4 August 2018
Problem
Find all prime number such that
and
are also prime.
Solution
If , then
and
. Since
is not prime,
can not be
. If
, then
and
. Both of the numbers are prime, so
can be
.
The rest of the prime numbers are congruent to ,
,
, and
modulo
, so
is congruent to
or
modulo
. If
, then
. If
, then
. That means if
is congruent to
,
,
, or
modulo
, then either
or
can be written in the form
.
The only way for to equal
is when
or
, which are not prime numbers. Thus, the rest of the primes can not result in
and
both prime, so the only solution is
.
See Also
2002 Indonesia MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 | Followed by Problem 7 |
All Indonesia MO Problems and Solutions |