The internal structure of natural numbers and one method for the definition of large prime numbers

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1 The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract It holds that every product of natural numbers can also be wrtten as a sum. The nverse does not hold when s excluded from the product. For ths reason the nvestgaton of natural numbers should be done through ther sum and not through ther product. Such an nvestgaton s presented n the present artcle. We prove that prmes play the same role for odd numbers as the powers of for even numbers and vce versa. The followng theorem s proven: Every natural number except for 0 and can be unquely wrtten as a lnear combnaton of consecutve powers of wth the coeffcents of the lnear combnaton beng - or +. Ths theorem reveals a set of symmetres n the nternal order of natural numbers whch cannot be derved when studyng natural numbers on the bass of the product. From such a symmetry a method for dentfyng large prme numbers s derved. MSC classfcatons: A4 N05 Keywords: Prme numbers composte numbers. INTRODUCTION It holds that every product of natural numbers can also be wrtten as a sum. The nverse (.e. each sum of natural numbers can be wrtten as a product) does not hold when s excluded from the product. Ths s due to prme numbers whch can be wrtten as a product only n the form of p p p. For ths reason the nvestgaton of natural numbers should be done through ther sum and not through ther product. Such an nvestgaton s presented n the present artcle. We prove that each natural number can be wrtten as a sum of three or more consecutve natural numbers except of the powers of and the prme numbers. Each power of and each prme number cannot be wrtten as a sum of three or more consecutve natural numbers. Prmes play the same role for odd numbers as the powers of for even numbers and vce versa. We prove a theorem whch s analogous to the fundamental theorem of arthmetc when we study the postve ntegers wth respect to addton: Every natural number wth the excepton of 0 and can be wrtten n a unque way as a lnear combnaton of consecutve powers of wth the coeffcents of the lnear combnaton beng - or +. Ths theorem reveals a set of symmetres n the nternal order of natural numbers whch cannot be derved when studyng natural numbers on the bass of the product. From such a symmetry a method for dentfyng large prme numbers s derved.

2 . THE SEQUENCE μkn We consder the sequence of natural numbers k n k k k... k n k na For the sequence Theorem.. For the sequence Proof. kn kn. kn. n k n the followng theorem holds: the followng hold:. (.). No element of the sequence s a prme number. 3. No element of the sequence s a power of. 4. The range of the sequence s all natural numbers that are not prmes and are not powers of. kn as a sum of natural numbers... na34... and therefore t holds that n n 3. Also we have that kn4 k n 3 snce k and na Thus the product n k n kn s always a product of two natural numbers dfferent than thus the natural number kn cannot be prme.

3 3. Let that the natural number kn such as n k n s a power of. Then t exsts n k n n k n. (.) Equaton (.) can hold f and only f there exst n k n and equvalently such as n. (.3) n k We elmnate n from equatons (.3) and we obtan k and equvalently k whch s mpossble snce the frst part of the equaton s an odd number and the second part s an even number. Thus the range of the sequence kn 4. We now prove that the range of the sequence kn does not nclude the powers of ncludes all natural numbers that are not prmes and are not powers of. Let a random natural number N whch s not a prme nor a power of. Then N can be wrtten n the form N 3. Let where at least one of the s an odd number be an odd number prove that there are always exst k and na34... such as N k n. We consder the followng two pars of k and n : 3. We wll. k k n n (.4) k k. (.5) n n 3

4 For every t holds ether the nequalty or the nequalty Thus for each par of naturals where s odd at least one of the pars k n k n equaton (.4) s holds that of equatons (.4) (.5) s defned. We now prove that when the natural number n k 0.. For then the natural number k 0 and from equatons (.5) we have that k n and because k n we obtan 3. k from equatons (.4) we take of equaton (.5) s k. k of and addtonally t We now prove that when k 0 n equatons (.5) then n equatons (.4) t s k and n. For k 0 from equatons (.5) we obtan and from equatons (.4) we get k. n We now prove that at least one of the k 0 k 0. k and Then from equatons (.4) and (.5) we have that k s postve. Let 0 0. (.6) Takng nto account that s odd that s we obtan from nequaltes (.6) whch s absurd. Thus at least one of For equatons (.4) we take k and k s postve. 4

5 n k n k n. N For equatons (.5) we obtan n k n k n. N Thus there are always exst and na34... such as k N k n for every N whch s not a prme number and s not a power of. Example.. For the natural number N and from equatons (.4) we get 6 5 k k 6 n n 5 4 thus we obtan N 40 we have Example.. For the natural number N 5 N there are two cases. Frst case: N and from equatons (.4) we obtan 34 3 k k 6 n n 3 thus 5 6. Second case: 5

6 N and from equatons (.5) we obtan 7 6 k k 6 n n 6 5 thus The second example expresses a general property of the sequence kn. The more composte an odd number that s not prme (or an even number that s not a power of ) s the more are the Example.3. kn combnatons that generate t a b c In the transtve property of multplcaton when wrtng a composte odd number or an even number that s not a power of as a product of two natural numbers we use the same natural numbers :. On the contrary the natural number kn usng dfferent natural numbers k and na through equatons (.4) (.5). Ths dfference between the can be wrtten n the form product and the sum can also become evdent n example.3:

7 From Theorem. the followng corollary s derved: Corollary... Every natural number whch s not a power of and s not a prme can be wrtten as the sum of three or more consecutve natural numbers.. Every power of and every prme number cannot be wrtten as the sum of three or more consecutve natural numbers. Proof. Corollary. s a drect consequence of Theorem.. 3. THE CONCEPT OF REARRANGEMENT In ths paragraph we present the concept of rearrangement of the composte odd numbers and even numbers that are not power of. Moreover we prove some of the consequences of the rearrangement n the Dophantne analyss. The concept of rearrangement s gven from the followng defnton: Defnton. We say that the sequence k n k n A natural numbers k n A k n k n k n k n s rearranged f there exst such as. (3.) From equaton (.) wrtten n the form of k n k k k... k n two dfferent types of rearrangement are derved: The compresson durng whch n decreases wth a smultaneous ncrease of k. The «decompresson» durng whch n ncreases wth a smultaneous decrease of k. The followng theorem provdes the crteron for the rearrangement of the sequence kn. Theorem 3... The sequence k n k n k n k n A can be compressed (3.) f and only f there exst n whch satsfes the equaton k n n 0 n. The sequence k n k n k n. (3.3) k n A can be decompressed (3.4) 7

8 f and only f there exst k whch satsfes the equaton k n n 0 k. (3.5) 3. The odd number s prme f and only f the sequence l kn l k na (3.6) cannot be rearranged. 4. The odd s prme f and only f the sequence (3.7) cannot be rearranged. Proof.. We prove part of the corollary and smlarly number can also be proven. From equaton (4.) we conclude that the sequence k n such as k n k n. In ths equaton the natural number n can be compressed f and only f there exst belongs to the set A and thus n n. Next from equatons (.) we obtan k n k n n k n n k n and after the calculatons we get equaton (3.3). 3. The sequence (3.6) s derved from equatons (.4) or (.5) for and l. Thus n the product the only odd number s. If the sequence kn n equaton (3.6) cannot be rearranged then the odd number has no dvsors. Thus s prme. Obvously the nverse also holds. 4. Frst we prove equatons (3.7). From equaton (.) we obtan: 8

9 . In case that the odd number s prme n equatons (.4) (.5) the natural numbers and from equaton (.5) we get k n. Thus the sequence kn cannot be rearranged. Conversely f the sequence thus s prme. We now prove the followng corollary: Corollary 3... The odd number are unque cannot be rearranged the odd number cannot be composte and odd s decompressed and compressed f and only f the odd number. The even number l l odd l 3 l l s composte. (3.9) (3.8) cannot be decompressed whle t compresses f and only f the odd number s composte. 3. The even number odd l l l l l (3.0) cannot be compressed whle t decompresses f and only f the odd number s composte. 9

10 4. Every even number that s not a power of can be wrtten ether n the form of equaton (3.9) or n the form of equaton (3.0). Proof.. It s derved drectly through number (4) of Theorem 3.. A second proof can be derved through equatons (.4) (5) snce every composte odd can be wrtten n the form of odds. 3. Let the even number l odd l. (3.) From equaton (.4) we obtan l l k n and snce k n k n we get (3.) l and equvalently l 3. In the second of equatons (3.) the natural number n obtans the maxmum possble value of n and thus the natural number k takes the mnmum possble value n the frst of equatons (3.). Thus the even number l cannot decompress. If the odd number s composte then t can be wrtten n the form of odds l. Therefore the natural number l decompresses snce from equatons (3.) t can be wrtten n the form of kn n. Smlarly the proof of 3 s derved from equatons (.5). wth 4. From the above proof process t follows that every even number that s not a power of can be wrtten ether n the form of equaton (3.9) or n the form of equaton (3.0). 0

11 By substtutng P prme n equatons of Theorem 3. and of corollary 3. four sets of equatons are derved each ncludng nfnte mpossble dophantne equatons. Example 3.. The odd number P s prme. Thus combnng () of Theorem 3. wth () of corollary 3. we conclude that there s no par wth whch satsfes the dophantne equaton We now prove the followng corollary: Corollary 3. The square of every prme number can be unquely wrtten as the sum of consecutve natural numbers. Proof. For P prme n equaton (3.5) we obtan P P P. (3.3) P Accordng wth 4 of Theorem 3. the odd cannot be rearranged. Thus the odd can be unquely wrtten as the sum of consecutve natural numbers as gven from equaton (3.3). Example 3.. The odd P 7 s prme. From equaton (3.3) for P 7 we obtan and from equaton (.) we get whch s the only way n whch the odd number 89 numbers. can be wrtten as a sum of consecutve natural 4. NATURAL NUMBERS AS LINEAR COMBINATION OF CONSECUTIVE POWERS OF Accordng to the fundamental theorem of arthmetc every natural number can be unquely wrtten as a product of powers of prme numbers. The prevously presented study reveals a correspondence between odd prme numbers and the powers of. Thus the queston arses whether there exsts a theorem for the powers of correspondng to the fundamental theorem of arthmetc. The answer s gven by the followng theorem: Theorem 4.. Every natural number wth the excepton of 0 and can be unquely wrtten as a lnear combnaton of consecutve powers of wth the coeffcents of the lnear combnaton beng - or +. Proof. Let the odd number as gven from equaton

12 From equaton (4.) for we obtan. (4.) We now examne the case where can obtan s. The lowest value that the odd number of equaton (4.) mn... mn. (4.) The largest value that the odd number of equaton (4.) can obtan s max... max. (4.3) Thus for the odd numbers mn of equaton (4.) the followng nequalty holds. (4.4) The number N N N max of odd numbers n the closed nterval s max mn. (4.5) The ntegers 0... n equaton (4.) can take only two values thus equaton (4.) gves exactly N odd numbers. Therefore for every equaton (4.) gves all odd numbers n the nterval. We now prove the theorem for the even numbers. Every even number whch s a power of can be unquely wrtten n the form of. We now consder the case where the even number of s not a power of. In that case accordng to corollary 3. the even number s wrtten n the form odd l l. (4.6)

13 We now prove that the even number can be unquely wrtten n the form of equaton (4.6). If we assume that the even number can be wrtten n the form of ' ' l l l l ' ' ' ' l l ' ll ( ) odd (4.7) the we obtan ' ll ' l l ' ' whch s mpossble snce the frst part of ths equaton s even and the second odd. Thus t s and we take that from equaton (4.7). Therefore every even number that s not a power of can be unquely wrtten n the form of equaton (4.6). The odd number of equaton (4.6) can be unquely wrtten n the form of equaton (4.) thus from equaton (4.6) t s derved that every even number that s not a power of can be unquely wrtten n the form of equaton ' l l ' l l l (4.8) and equvalently l l l 0 l l 0... For we take 0 0. (4.9) thus t can be wrtten n two ways n the form of equaton (4.). Both the odds of equaton (4.) and the evens of the equaton (4.8) are postve. Thus 0 cannot be wrtten ether n the form of equaton (4.) or n the form of equaton (4.8). In order to wrte an odd number 3 from nequalty (4.4). Then we calculate the sum n the form of equaton (4.) we ntally defne the 3

14 . If t holds that we add the whereas f t holds that then we subtract t. By repeatng the process exactly tmes we wrte the odd number n the form of equaton (4.). The number of steps needed n order to wrte the odd number n the form of equaton (4.) s extremely low compared to the magntude of the odd number as derved from nequalty (4.4). Example 4.. For the odd number 3 we obtan from nequalty (4.4) 3 4 thus 3. Then we have (thus (thus s added) s subtracted) (thus 0 s added) mn F s Fermat numbers F s can be wrtten drectly n the form of equaton (4.) snce they are of the form s s s s s s 3 mn.... (4.0) s Mersenne numbers max M p can be wrtten drectly n the form of equaton (4.) snce they are of the form M p 3 p p max p p p.... (4.) p prme In order to wrte an even number that s not a power of n the form of equaton (4.) ntally t s consecutvely dvded by and t takes of the form of equaton (4.6). Then we wrte the odd number n the form of equaton (4.). 4 Example 4.. By consecutvely dvdng the even number 368 by we obtan Then we wrte the odd number 3 n the form of equaton (4.) 3 and we get

15 Ths equaton gves the unque way n whch the even number 368 can be wrtten n the form of equaton (4.9). From nequalty (4.4) we obtan log log log from whch we get log log log log and fnally log log (4.) where log log the nteger part of log. log We now gve the followng defnton: Defnton 4.. We defne as the conjugate of the odd (4.3) the odd j j j 0... j0 j j (4.4) for whch t holds (4.5) k k k For conjugate odds the followng corollary holds: Corollary 4.. For the conjugate odds and the followng hold: 5

16 .. (4.6). 3. (4.7) 3. s dvsble by 3 f and only f s dvsble by 3. Proof..The of the corollary s an mmedate consequence of defnton 4... From equatons (4.3) (4.4) and (4.5) we get and equvalently If the odd s dvsble by 3 then t s wrtten n the form 3 x x odd and from equaton (4.7) we get 3x 3 and equvalently 3 x. Smlarly we can prove the nverse. 5. THE HARMONIC ODD NUMBERS AND A METHOD FOR DEFINING LARGE PRIME NUMBERS The harmonc symmetry: We defne as harmonc the odd numbers of equaton (4.) for whch the sgns of alternate: From equatons (5.) (5.) and defnton 4. we obtan. (5.). (5.) 3 (5.3) for the par of harmonc odd numbers. A method for the determnaton of large prme numbers emerges from the study we presented. Ths method s completely dfferent from prevous methods [-].When we consder the prme factorzaton of the odd ntegers... (5.4) 6

17 ... (5.5) 3 (5.6) we have the followng statement: The factors of ether or from the factorzaton of and consst of a set of small prme factors and one large factor. Hence Followng are examples where we have chosen arbtrary even (5.4) (5.5) of equatons (5.4) (5.5) we get a large prme number n equatons

18 s prme ? ?? Equatons (5.4) (5.5) and (5.6) are a specal case of equatons (5.9) for 0. The general equatons (5.7) (5.8) and (5.9) gve all possble varatons of the method. For example for 838 a value of that dd not gve a large prme number n the prevous examples from equaton (5.7) for we get (5.7) (5.8) 8

19 For 66 and from equaton (5.7) we get = s prme

20 s prme s prme Theorem 4. hghlghts addtonal symmetres of the nternal structure of the natural numbers. We wll not expand upon these symmetres n the current artcle. 0

21 References. Apostol Tom M. Introducton to analytc number theory. Sprnger Scence & Busness Meda 03.. Mann Yu I. and Alexe A. Panchshkn. Number theory I: fundamental problems deas and theores. Vol. 49. Sprnger Scence & Busness Meda Damond Harold G. "Elementary methods n the study of the dstrbuton of prme numbers." Bulletn of the Amercan Mathematcal Socety 7.3 (98): Newman Davd J. "Smple analytc proof of the prme number theorem." The Amercan Mathematcal Monthly 87.9 (980): Ttchmarsh Edward Charles and Davd Rodney Heath-Brown. The theory of the Remann zetafuncton. Oxford Unversty Press Poussn Charles Jean de La Vallee. Sur la foncton [zeta](s) de Remann et le nombre des nombres premers nfereurs à une lmte donnée. Vol. 59. Hayez Perera N. Costa. "A Short Proof of Chebyshev's Theorem." The Amercan Mathematcal Monthly 9.7 (985): Bateman P. T. J. L. Selfrdge and S. Samuel Wagstaff. "The Edtor's Corner: The New Mersenne Conjecture." The Amercan Mathematcal Monthly 96. (989): Deléglse Marc and Joël Rvat. "Computng the summaton of the Möbus functon." Expermental Mathematcs 5.4 (996): Cashwell Edmond and C. J. Everett. "The rng of number-theoretc functons." Pacfc Journal of Mathematcs 9.4 (959): Abramowtz Mlton Irene A. Stegun and Robert H. Romer. "Handbook of mathematcal functons wth formulas graphs and mathematcal tables." (988):