Twin Prime Conjecture
The Twin Prime Conjecture is a conjecture (i.e., not a theorem) that states that there are infinitely many pairs of twin primes, i.e. pairs of primes that differ by .
Contents
[hide]Failed Proofs
Using an infinite series
One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of Dirichlet's Theorem. If one can show that the sum
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of the reciprocals of twin primes diverges, this would imply that there are infinitely many twin primes. Unfortunately, it has been shown that this sum converges to a constant , known as Brun's constant. This could mean either that there are finitely many twin prime pairs or that they are spaced "too far apart" for that series to diverge.
Yitang Zhang approach
A weaker version of twin prime conjecture was proved by Yitang Zhang in 2013. This version stated that there are infinitely many pairs of primes that differ by a finite number. The number Yitang chose was 7,000,000. Terence Tao and other people has reduced that boundary to 246 more numbers.
Elementary proof
Let be the multiplication of the first s prime numbers.
Let
be the sth prime number
Let
be the set of numbers relatively prime to
and less than
.
and
where
in
and
and
Pair up numbers generated from two arithmetic progression where
If it is not possible to generate a non-prime in each pair then there exist a twin prime.
The base case for numbers which differ by in
is
and
. Induction there will always be two numbers which differ by
in
.
Let
and
will propagate
pairs of elements in
which differ by
where
and
and
because only the unique values
and
in their respective arithmetic progression has the factor of
when
and
will propagate
pairs of elements in
which differ by
where
and
and
because only the unique values
and
in their respective arithmetic progression has the factor of
when
All non-primes numbers generated by where
in
and
can also be found in
Therefore removing all numbers from
with odd factors between and including
to
will either leave an empty set or a set only containing prime numbers.
Using the fact that there is a fix set of sequential numbers between numbers with the same factor f in arithmetic progression.
where
and
is an unique pick from
if
is a factor of a number in
then there is an unique value in
which
is a factor when
,
Mark possible non-prime in pair values generated from arithmetic progression and
where values are paired if
.
The largest factor to eliminate is smaller than the number of pairs elements generate by two arithmetic progressions in
where
and
in
Can guarantee there are
elements without the factor of
in a consecutive sequence of
elements from arithmetic progression
where the two numbers with factor of
are generated in two different arithmetic progression in two different pairs. Assume the remaining
pairs without factor of
are in a consecutive sequence eliminate the next smaller odd number which differs by
.
Assume the remaining
pairs without factor of
are in a consecutive sequence eliminate the next smaller odd number which differs by
.
Repeat until the number of elements in consecutive sequence is
. Removing numbers with factor of
. There must be a pair of numbers where both of them are prime numbers.
There must be infinite number of twin primes.
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