VIEWS: 6 PAGES: 6 POSTED ON: 10/26/2011 Public Domain
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.