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