proof_of_twin_prime-149367 by panniuniu

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									ELEMENTARY PROOF OF THE TWIN PRIME CONJECTURE
   USING THE LITTLE KNOWN SUNDARAM’S SIEVE
           Author: Mr. Ayodeji Awojobi

Prime Numbers (or Primes)
A prime number (or prime) is defined here as a positive integer number
that is divisible by only 2 positive integer numbers which are 1 and the
prime number itself. In short, a prime number has only 2 factors, 1 and
the prime number itself. The first few prime numbers are
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,..
It is a simple matter to prove that there are infinitely many primes even
though they slowly ‘thin out’, i.e. they occur less frequently as one
searches for them further down the number line. Euclid, more than 2000
years ago, proved the infinitude of primes using a thought experiment. As
can be seen, 2 is the only even prime number and 1 doesn’t qualify as a
prime number because it has only one factor which is 1. It is not difficult
to show that all prime numbers apart from 2 and 3 are of the form 6n+1
or 6n-1, where n is a positive integer. Thus they are numbers that are
one more and/or one less than some multiples of 6. It has been proved
that there are approximately N/lnN primes less than N, where N is a
positive integer and lnN is the natural logarithm of N. This is called the
Prime Number Theorem.

Twin Primes
The twin prime conjecture simply states that there are an infinite number
of twin primes. Twin primes are primes that have a difference of 2
e.g. 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61.
No matter how far down the positive number line one searches, the twin
primes seem to ‘pop up’ every now and then without abating, hence the
conjecture. As can be seen and as previously implied, twin primes occur
one more and one less than some multiples of 6.

Sundaram’s Sieve
It was whilst reading an article titled, ‘Sundaram’s Sieve’, by Julian Havil
in the March 2009 edition of the Plus Magazine that the author had a
‘eureka moment’ and intuitively knew that this sieve is the key to proving
the age old problem of the twin prime conjecture. The beauty of
Sundaram’s sieve compared to the sieve of Eratosthenes is two-fold.
Firstly, it doesn’t sieve out the primes, it sieves out the odd non-primes.
Secondly, it doesn’t require a prime to sieve out other primes as the sieve
of Eratosthenes does require. The sieve is named after an obscure East
Indian mathematician by the name of S.P. Sundaram, who discovered it
in the 1930s. The sieve is an infinite number of infinite arithmetic number
sequences. Part of the sieve is shown here.

      4       7      10      13      16      19      22      25      28
      7      12      17      22      27      32      37      42      47
     10      17      24      31      38      45      52      59      66
     13      22      31      40      49      58      67      76      85
     16      27      38      49      60      71      82      93     104
     19      32      45      58      71      84      97     110     123
     22      37      52      67      82      97     112     127     142
     25      42      59      76      93     110     127     144     161
     28      47      66      85     104     123     142     161     180
     31      52      73      94     115     136     157     178     199
     34      57      80     103     126     149     172     195     218
     37      62      87     112     137     162     187     212     237
     40      67      94     121     148     175     202     229     256
     43      72     101     130     159     188     217     246     275
     46      77     108     139     170     201     232     263     294
     49      82     115     148     181     214     247     280     313
     52      87     122     157     192     227     262     297     332
     55      92     129     166     203     240     277     314     351
     58      97     136     175     214     253     292     331     370
     61     102     143     184     225     266     307     348     389
     64     107     150     193     236     279     322     365     408
     67     112     157     202     247     292     337     382     427
     70     117     164     211     258     305     352     399     446
     73     122     171     220     269     318     367     416     465
     76     127     178     229     280     331     382     433     484
     79     132     185     238     291     344     397     450     503
     82     137     192     247     302     357     412     467     522
     85     142     199     256     313     370     427     484     541
     88     147     206     265     324     383     442     501     560
     91     152     213     274     335     396     457     518     579
     94     157     220     283     346     409     472     535     598
     97     162     227     292     357     422     487     552     617
    100     167     234     301     368     435     502     569     636
    103     172     241     310     379     448     517     586     655

The sieve shows a variety of patterns including the fact that the numbers
in the first row are the same as the numbers in the first column,the
numbers in the second row are the same as the numbers in the second
column etc. Also, the difference between consecutive numbers in each
row (or column) increases by 2 each time for the next row (or column). It
is very remarkable and yet not difficult to prove that doubling any
number appearing in the sieve and then adding 1 to the answer results in
a non – prime odd number. A lot of numbers are repeated in the sieve. In
fact the numbers appearing in some rows (or columns) are all repetitions
of numbers appearing in some other rows (or columns). For instance the
7th,12th,17th,22nd,27th, 32nd ,…….rows (or columns), which differ by 5, will
have numbers that all appear in the second row (or column). Elementary
algebra can show why this is the case. Notice also that the numbers in
the second row (or column) will sieve out all the multiples of 5 greater
than 5. For example the first number in the second row (or column) is 7
and 7 x 2 + 1 = 15, the second number is 12 and 12 x 2 + 1 = 25. Notice
also that the first row (or column) produces the odd non-primes that are
not of the form 6n+1 or 6n-1, e.g. 4 x 2 + 1= 9, 7 x 2 + 1 = 15, 10 x 2
+ 1 = 21 for the first 3 numbers in this row (or column). Notice that the
4th, 7th, 10th, 13th, 16th, 19th, 22nd,…….rows (or columns), which differ by
3, will have numbers that all appear in the first row (or column).
Elementary algebra again shows why this happens. Sundaram’s sieve can
now be modified by removing the rows that produce multiples of 5
greater than 5 and odd non-primes that aren’t of the form 6n+1 or 6n-1.
Thus the modified Sundaram’s sieve will only use the following rows.

3rd ,5th,6th,8th,9th,11th,14th,15th,18th,20th,21st,23rd,24th,26th,29th,30th,33rd,…

   2   1   2   1   2   3    1    3    2    1   2     1    2    3    1     3

The numbers immediately above show the difference between
consecutive rows in the modified sieve and a pattern can clearly be seen.
It is not difficult to show that the general equation for producing a
number in Sundaram’s sieve in row x and column y is 2xy + x + y. The x
and y values of 3,5,6,8,9,11,14,15,18,20,21,23,24,26,29,30,33,…… will
therefore produce odd non-primes of the form 6n+1 or 6n-1, none of
which will be a multiple of 5. Of course repetitions of numbers in the
modified sieve will still occur (and not only when x=y). In fact the
numbers in further rows will still replicate the numbers in previous rows
in the modified sieve. For instance the numbers in the 24th row can all be
found in the 3rd row.

Mechanism of how prime numbers are created using the sieve
It is not difficult to show that of the N/6 multiples of 6 less than N, there
are 2/15 N multiples of 6 less than N (N is a positive integer) that qualify
to have a chance of harboring twin primes in their vicinity i.e. these
multiples of 6 do not have multiples of 5 that are one more or one less
than them. Thus, some of the 2/15 N potential twin prime number
positions on the number line will end up harboring twin primes. The
thought experiment imagines a situation where the twin primes exist
amongst the non-twin primes initially. The first row of the modified
Sundaram’s sieve, i.e. the 3rd row of Sundaram’s sieve, is then used to
eliminate a fraction of the 2/15 N potential twin primes by making use of
the 3rd, 5th, 6th, 8th, 9th, 11th, …. columns, as discussed earlier. Thus the
following numbers will be eliminated, 24 x 2 + 1 = 49, 38 x 2 + 1 = 77,
45 x 2 + 1 = 91, 59 x 2 + 1 = 119, 66 x 2 + 1 = 133,……..and thus seal
the fate of some of the potential twin prime positions, i.e. these positions
cannot harbor twin primes. One can imagine this first row of the modified
Sundaram’s sieve as a wave with fluctuating wavelength taking out a
fraction of the 2/15 N potential twin prime positions. Thus, 8 hops out of a
possible 15 hops per cycle is made in order to avoid multiples of 5 and
odd non-primes that are not of the form 6n+1 or 6n-1 (see previous sub
heading). The next row (or wave) of the modified Sundaram’s sieve does
something similar and will eliminate more potential twin primes by
making new ‘hits’. However, some numbers that were ‘hit’ by the first
wave will be ‘hit’ again and also numbers that are 2 more or 2 less than
numbers that were previously ‘hit’, will be ‘hit’. The last phrase indicates
a kind of repetition because the potential twin prime position has already
been ‘hit’ by the first wave. This process carries on for wave after wave of
the modified Sundaram’s sieve. The proportion of repetitions will keep on
increasing in number and the proportion of new ‘hits’ will keep on
decreasing in number. It should be borne in mind that any 2 waves
interacting with each other will always produce numbers that are
repeated at equal intervals. Also they will also produce numbers that
differ by 2 repeated at equal intervals (apart from those rows where all
the numbers in one row can be found in the other as discussed
previously). Simple algebra can show why this is the case. Thus, imagine
a situation where a wave has its turn on an infinite number line that
harbors only the 2/15 N potential twin prime positions. The finite part of
the number line to the left of the starting point of this wave is now fixed
in time (the numbers that weren’t ‘hit’ are primes) and no subsequent
waves can affect it. To the right of this starting point of the wave there
will be repetitive patterns of hits and non-hits all the way down the
infinite number line based on discussions directly above. Basically the
proportion of ‘hits’ and ‘non hits’ stays the same in this part of the
number line. Waves coming after will alter this proportion continually.

Proof of the twin prime conjecture using a thought experiment
The scene is now set to prove that the twin prime conjecture is true i.e.
there are an infinite number of twin primes. It is an elementary proof with
no rigorous mathematics involved. As previously mentioned, Euclid
proved the infinitude of primes using a thought experiment and the proof
of the twin prime conjecture will now be made using a thought
experiment and the modified Sundaram’s sieve. The first question that
can be asked is, ‘Will there be a point where no new ‘hits’ will be made’?
The obvious answer is ‘no’ because this will cause the PNT to be negated
since the proportion of primes from this point on will remain static and
wouldn’t ‘thin out’. The next question is, ‘Will there come a point far down
the infinite number line where all potential twin prime positions would
have been wiped out’? The answer to this question is ‘no’ due to the
following reasons. These reasons would mean that the twin prime
conjecture is true. The infinitude of primes makes it very clear that the
infinite number of waves of the modified Sundaram’s sieve will not ‘hit’ all
the infinite number of potential prime positions i.e. numbers of the form
6n+1 or 6n-1 that aren’t multiples of 5. This means that a large fraction
of the infinite numbers will be ‘hit’ and a smaller fraction will not be ‘hit’,
i.e. the primes. Of the fraction that will not be ‘hit’, twin primes will be
amongst them for the simple reason that no wave can ‘take out’ all the
remaining infinite potential twin primes. The wave can only ‘take out’ a
fraction of the remaining potential twin primes. One can ask how the last
statement can be justified. It should be clear from previous discussions
that the waves are generated by arithmetic number sequences and when
they begin to interact and interfere with one another they leave behind
prime numbers that do not form arithmetic number sequences. Therefore
no one arithmetic number sequence can wipe them all out, just a fraction
can be removed. This therefore means that there are infinitely many
potential twin primes which will never run out. Some of these would be
‘hit’ on either one of its 6n+1 or 6n-1 positions to leave a prime, some
could be ‘hit’ by more than one wave on both the 6n+1 and 6n-1
positions to leave no prime and some would not be ‘hit’ to leave an
infinite number of twin primes.

								
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