Being the Goldbach Conjecture considered the most famous conjecture unsolved, concerning prime numbers, this paper is dedicated to prime numbers, seeking to create a method to determine prime pairs for a given even number. This method, described in the paper, allows to determine a Goldbach partition for an even number, sometimes even all, in a practical and effective way, without using complex algorithms

Dias, Margarida Raposo

Garrão, Ana Paula

Sousa, José Emanuel

English

European Scientific Journal (European Scientific Institute)

European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 497 GOLDBACH CONJECTURE: METHOD TO DETERMINE PRIME PAIRS    José Emanuel Sousa, MA EBI Rabo de Peixe, Portugal Ana Paula Garrão, PhD Margarida Raposo Dias, PhD Universidade dos Açores, Departamento de Matemática, Portugal Núcleo Interdisciplinar da Criança e do Adolescente da Universidade dos Açores, Portugal   Abstract   Being the Goldbach Conjecture considered the most famous conjecture unsolved, concerning prime numbers, this paper is dedicated to prime numbers, seeking to create a method to determine prime pairs for a given even number. This method, described in the paper, allows to determine a Goldbach partition for an even number, sometimes even all, in a practical and effective way, without using complex algorithms.      Keywords: Goldbach Conjecture; prime numbers; even number; Goldbach partition     Introduction  Goldbach Conjecture is a topic that had always aroused curiosity through the centuries for its apparent simplicity. However, 273 years after its first appearance in a letter sent by Christian Goldbach to Leonhard Euler in June 1742, the conjecture has not yet been proved. Neither Goldbach nor Euler could prove it, and as them, many others tried, with no result. To be fair, mainly in the last century, Number Theory has evolved due to many mathematicians, whom, in seeking to prove Goldbach Conjecture, were able to grant results also important in Number Theory.   In this paper is described a method to determine Goldbach partitions of an even number, based on the concept of prime number and prime factors.  Goldbach Partition Goldbach Conjecture is one of the oldest problems in number theory and in all mathematics. It states:  European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 498 “Every even integer greater than 2 can be expressed as the sum of two prime numbers”. In other words: Let a  be an even number greater than 2. Then, there exist p , prime number, such that a p  is a prime number. The pair ( p , a p ) is called a Goldbach partition of a .  The following are examples of Goldbach partitions for some even numbers:   4 2 26 3 38 3 524 5 19 7 17 11 13...          The method we are going to present in this paper is based on a well-known primality test and in the prime factorization of an integer. Proposition 1: If a positive integer 1a   has no prime divisor lower or equal to a , then a  is prime number.  Proof: By contraposition we must prove that: if a positive integer 1a   is composite then a admits at least one prime factor p a .   Let 1a   be a composite number, then it is possible to write a bc , for some positive integers b and c , both higher than 1. Suppose, without loss of generality, that b c . Then,  2b bc a b a      Since 1b  , the Fundamental Theorem of Arithmetic assures that bhas at least one prime divisor p , with p b a  . Given that p divides b andb divides a , we must conclude that p  divides a , i.e., p a  is a prime divisor of a . Thus, a  admits at least one prime factor lower than a .                                      Every positive integer 2a   can be written, in exactly one way, as a product of prime powers, 1 21 2 ...nss sna p p p      (1)  where ip  is a prime factor, is  a natural number, for  1,2,...,i n , and 1 2 ... np p p   , by convention. This factorization is called prime decomposition of a. European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 499 Remark: Notice that, given a positive integer 2a  , with prime decomposition defined by (1), for any prime number p , such that ip p , {1,  2,  3,  ,  }i n   and 1p a  , we have that a p  is not divisible by ip , {1,  2,  3,  ,  }i n  . In fact, suppose, by reduction to the absurd, that exists ip  such that ip  divides a p , then ip  divides ( ( ))a a p  , that is ip  divides p, which is an absurd, since p  and ip  are distinct prime numbers.   Example 1: Consider 30 2 3 5a     . Calculating a p , for each prime number 29p  , distinct of 2, 3 and 5, we have: 30 7 2330 11 1930 13 1730 17 1330 19 1130 23 7        We observe that a p  is not divisible by 2, 3 or 5.   In this case, a p  always results in a prime.   Example 2: Consider 70 2 5 7a     . Calculating a p , for each prime number 69p  , distinct of 2, 5 and 7, we have: 70 3 6770 11 5970 13 57...70 67 3      We observe that a p  is not divisible by 2, 5 or 7.  Moreover, we can verify that, in this case, a p  is a prime number or multiple of 3.  The next result is the foundation of the method to determine Goldbach partitions.   Proposition 2: Let 8a   be an even number, which prime decomposition is written as, 1 21 2 ... ,nss sna p p p  with ip  prime, i , 1 2 ... np p p   , is    (2)  and let A  and B  be sets defined by1 2 1{ , ,..., } { : }n nA p p p and B t prime p t p t A      . If exists a prime number p , with p A , such that 21, 3np a p a      and, for each prime t B ,  a p  is not divisible by t , then a p  is prime. European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 500 Proof: Let p  be a prime number in the conditions defined.   The upper limit, 3a  , only limits superiorly the value of p , since if 3p a  , then a p  assumes the lower possible value, 3. For the lower limit, we have:  2 21 1 1n n np a p a p p a p p            So, a p  will be a prime number if a p  is not divisible by any prime number lower or equal to 1na p p   .  Since a p  is not divisible by ip A , {1,2,... }i n , as we already observed, and by hypothesis,  a p  is not divisible by any prime number t B , then a p  is not divisible by any prime number lower or equal to a p , leading to conclude, according to proposition 1, that a p  is necessarily a prime number.                In case the prime decomposition of a  include all prime numbers lower or equal to np , i.e., B  , then 3, 5, ..., na a a p    are divisible by 3, 5, …, np , respectively, then we can set the upper limit of the interval defined above to 1na p  .  We use this result to find Goldbach partitions of an even integer in the next examples.  Example 3: Consider 2220 2 5 11a     . We have  2,5,11A  , 3,7B  , 11np   and 1 13np   .  Therefore, 2 21 220 13 51; 3 217na p a      . In the interval  51,217 , we can find the following prime numbers: 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211  For each p in the previous list, a p  is: 167 161 159 153 149 147 141 137 131 123 119 117 113 111 107 93 89 83 81 71 69 63 57 53 47 41 39 29 27 23 21 9   Removing the multiples of 3 and 7, i.e., 161, 159, 153, 147, 141, 123, 119, 117, 111, 93, 81, 69, 63, 57, 39, 27, 21 and 9, the remaining numbers are necessarily prime numbers: 23, 29, 41, 47, 53, 71, 83, 89, 107, 113, 131, 137, 149 and 167. Therefore, the following pairs:  European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 501 (197,23) (191,29) (179,41) (173,47) (167,53) (149,71) (137,83) (131,89) (113,107) (107,113) (89,131) (83,137) (71,149) (53,167)  are Goldbach partitions for 220.   Notice that some partitions are repeated: (167,53), (149,71), (137,83), (131,89) and (113,107).  Clearly, in case 21 2ana p   , we must consider the interval12,a na p    , if B  , and  2 , 3a a  , if B  .  In this case, all partitions of 220 were found. However, often not all the partitions of an even number are found using the interval21, 3na p a    . Example 4: Consider 260 2 3 5a     . We have  2,3,5A  , B , 5np   and 1 7np   . Therefore, 2 21 60 7 11, 302naa p       and 7 53a   .  Since B   and 11 30 , the interval to consider is  30,53 , which enables to determine all partitions of 60: (31,29) (37,23) (41,19) (43,17) (47,13) (53,7) Example 5: Consider 2 23150 2 3 5 7a      . We have  2,3,5,7A  , B  , 7np   and 1 11np   .  Therefore, 2 21 3150 11 3029na p      and 1 3150 11 3139na p     .  In this case, since B  , all prime numbers in the interval  3029,3139  will give a Goldbach partition, and thus, the following pairs are Goldbach partitions of 3150: (3037,113) (3041,109) (3049,101) (3061,89) (3067,83) (3079,71) (3083,67) (3089,61) (3109,41) (3119,31) (3121,29) (3137,13)   Obviously, not all Goldbach partitions were found with the interval. For instance, 1571 and 1579 define a Goldbach partition for 3150.  In previous examples, the considered interval contained at least one prime number. In case the interval contains no prime numbers, we must set a new interval, with greater range, which contains at least one prime number p, such that a p  is also prime.  Proposition 3: Let 8a   be an even number, with prime decomposition defined by (2) , and consider the associated sets A  and B . Let also k . European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 502  If exists p  prime, with np p , such that21 1,n k n kp a p a p        , a p  is not divisible by 1 2, ,...,n n n kp p p    and, for each prime t B ,  a p  is not divisible by t , then a p  is prime.  Proof:  Analogous of Proposition 2.  Example 6: Consider 112048 2a   . We have  2A  , B  , 2np   and1 3np   . Therefore, 2 21 2048 3 2039na p      and 1 2048 3 2045na p     .  In the interval  2039,2045 , there are no prime numbers. So, according to the proposition above, one must define a new interval, with a bigger range, that contains prime numbers.   For example, assuming 1k  , we have 2 5np    and the new interval is  2023,2043 , which contains two prime numbers, 2029 and 2039. Then a p  is 19 or 9. Therefore, we can define two possible Goldbach partitions: (2029,19) and (2039,9). Since 19 is not divisible by 3, the pair (2029,19) is a Goldbach partition for 2048.  As we verified, in the examples presented previously, sometimes all Goldbach partitions of an even number are determined, while in others, only a few are determined.  Given an even number 4a   with prime decomposition defined by (2) , ( )g a is the number of Goldbach´s partitions determined by the interval 21] , 3]na p a   and ( )G a  is the number of  Goldbach´s partitions admitted by a . Example 7: Consider 2 2300 2 3 5a     . We have  2,3,5A  ,B  , 5np  , 1 7np    and therefore, the interval to consider is  251,293 .  The partitions determined by the interval are:  (257,43) (263,37) (269,31) (271,29) (277,23) (281,19) (283,17) (293,7)  The missing partitions are: (151,149) (163,137) (173,127) (191,109) (193,107) (197,103) (199,101) (211,89) (227,73) (229,71) (233,67) (239,61) (241,59)    The missing partitions correspond to the interval  150,251 .   In this case, 8 of 21 partitions were determined with the interval, so (300) 8g   and (300) 21G  . European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 503  Clearly, if 21 2ana p   , there is no warranty that all the partitions are determined, missing the correspondent to 212,a na p    .   However, it is possible to indicate the upper limit of the number of partitions an even number can admit. Indeed, setting ( )a  the number of prime numbers lower than a , ( ) #a A   represents the number of primes lower than a  that are not prime factors of a , therefore, exists ( ) #a A   prime numbers candidates to define a partition, which can determine, at most, ( ) #2a A  different partitions of a . Then, we have: ( ) #( ) ( )2a Ag a G a     (3)   The method we described, to determine Goldbach partitions of an even number, can be summarized in the following diagrams:   Figure 6 - Method to determine a Goldbach Partition for a given even number.    If no prime pair was found with the previous diagram:  Figure 7 -  Alternative interval to determine a Goldbach Partition.  European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 504  We will now present examples to illustrate the different paths in the diagrams. Example 8: Consider 2 2252 2 3 7a     . We have  2,3,7A  ,  5B  , 7np   and 1 11np   . Therefore, 2 21 252 11 131na p     , 3 252 3 249a      and 2521262 2a  .  Since 131 126 , the interval to consider is  131,249 , where we can find the following prime numbers: 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241   For each p in the previous list, a p  is: 115 113 103 101 95 89 85 79 73 71 61 59 55 53 41 29 25 23 19 13 11   Removing the multiples of 5, i.e., 115, 95, 85, 55 and 25, the remaining numbers are necessarily prime numbers: 11, 13, 19, 23, 29, 41, 53, 59, 61, 71, 73, 79, 89, 101, 103, and 113. Therefore, the following pairs: (139,113) (149,103) (151,101) (163,89) (173,79) (179,73) (181,71) (191,61) (193,59) (199,53) (211,41) (223,29) (229,23) (233,19) (239,13) (241,11)  are Goldbach partitions for 252.  Example 9: Consider 2308 2 7 11a     . We have  2,7,11A  ,  3,5B  , 11np    and 1 13np   . Therefore, 2 21 308 13 139na p     , 1542a  and 3 305a   .  In this case, since 139 154 , the interval to consider is [154,305] , where we can find the following prime numbers: 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293  For each p in the previous list, a p  is: 151 145 141 135 129 127 117 115 111 109 97 85 81 79 75 69 67 57 51 45 39 37 31 27 25 15   Removing the multiples of 3 and 5, i.e., 145, 141, 129, 135, 117, 115, 111, 85, 81, 75, 69, 57, 51, 45, 39, 27, 25 and 15, the remaining numbers are European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 505 necessarily prime numbers: 31, 37, 67, 79, 97, 109, 127, and 151. Therefore, the following pairs: (157,151) (181,127) (199,109) (211,97) (229,79) (241,67) (271,37) (277,31)  are Goldbach partitions for 308.  Example 10: Consider 30 2 3 5a     . We have  2,3,5A  , B  , 5np    and 1 7np   . Therefore, 2 21 30 7 19na p      , 152a  and 1 30 7 23na p     .  In this case, since 19 15  , the interval to consider is [15,23] , where we can find the following prime numbers: 17, 19 and 23. Therefore, a p  is: 13, 11 or 7. Since B  , a p  is prime for any p .Therefore, the following pairs: (17,13) (19,11) (23,7) are Goldbach partitions for 30.  Example 11: Consider 2390390 2 3 5 7 11 13a        . We have 2,3,5,7,11,13A  , B  , 13np   and 1 17np   . Therefore, 2 21 390390 17 390101na p     , 1951952a  and 1 390390 17 390373na p     .  In this case, since 390301 195195 , the interval to consider is [390101,390373], where we can find the following prime numbers: 390107 390109 390113 390119 390151 390157 390161 390191 390193 390199 390209 390211 390223 390263 390281 390289 390307 390323 390343 390347 390353 390359 390367 390373     For each p in the previous list, a p  is: 283 281 277 271 239 233 229 199 197 191 181 179 167 127 109 101 83 67 47 43 37 31 23 17  Since B  , a p  is prime for any p . Therefore, the following pairs: (390107,283) (390109,281) (390113,277) (390119,271) (390151,239 (390157,233) (390161,229) (390191,199) (390193,197) (390199,191) (390209,181) (390211,179) (390223,167) (390263,127) (390281,109) (390289,101) (390307,83) (390323,67) (390343,47) (390347,43) (390353,37) (390359,31) (390367,23) (390373,17)  are Goldbach partitions for 390390. European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 506 Example 12: Consider 12 5995328 2 3a    . We have  2,3A  , B  , 3np    and 1 5np   .   Therefore 2 21 995328 5 995303na p     , 4976642a  and 1 995328 5 995323na p     . Since 995328 497664 , the interval to consider is [995303,995323] , where is not possible to find a prime number. Assuming 2k  , we have 3 11np    and the new interval to consider is [995207,995317] . In this interval we can find the following prime numbers:    995219 995227 995237 995243 995273 995303  For each p in the previous list, a p  is: 109 101 91 85 55 25  Removing the multiples of 5 and 7, i.e., 91, 85, 55 and 25, the remaining numbers are necessarily prime numbers: 109 and 101. Therefore, the following pairs: (995219,109) (995227,101)  are Goldbach partitions for 995328.   Conclusion  At first contact with Goldbach Conjecture, it appears like a problem that is simple to verify. Euler, in response to a Goldbach letter, affirmed the conjecture was probably true, but he was unable to prove it at that point. Certainly, Euler realized, before Goldbach, the apparent simplicity inherent to the conjecture, and probably thought Goldbach or other mathematician would, sooner or later, prove the conjecture, since it was reasonably simple to find a partition for a given even number. Still, despite the apparent simplicity, trying to prove has been fruitless, for several reasons, being the most relevant, the unpredictability and irregularity of prime numbers. Thus, Goldbach Conjecture has a hidden complexity, increased because an even number, with its prime factors, hinders any attempt to create a method or an algorithm, or even an approach to the problem, that could allow someone to prove the conjecture. European Scientific Journal August 2015 /SPECIAL/ edition ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431 507  At last, the most important feature of the method described is that, for a given even number and a prime number lower to it, to verify if the difference is also a prime number, it is only necessary to verify if this difference is divisible by a set of prime numbers, which is a lot easier and faster than to verify if it is a prime number.                           References: Sousa, José E. Conjetura de Goldbach – Uma visão aritmética. Master thesis, Universidade dos Açores, Departamento de Matemática, 2013. Boldoni, M. W., Ciliberto, C., Piacentini Cattaneo, G.M. Elementary Number Theory, Cryptography and Codes. Springer, 2009.       

GOLDBACH CONJECTURE: METHOD TO DETERMINE PRIME PAIRS

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GOLDBACH CONJECTURE: METHOD TO DETERMINE PRIME PAIRS

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