The internal structure of natural numbers, one method for the definition of large prime numbers, and a factorization test

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1 Fal verso The teral structure of atural umbers oe method for the defto of large prme umbers ad a factorzato test Emmaul Maousos APM Isttute for the Advacemet of Physcs ad Mathematcs 3 Poulou str. 53 Athes Greece Abstract It holds that every product of atural umbers ca also be wrtte as a sum. The verse does ot hold whe s excluded from the product. For ths reaso the vestgato of atural umbers should be doe through ther sum ad ot through ther product. Such a vestgato s preseted the preset artcle. We prove that prmes play the same role for odd umbers as the powers of for eve umbers ad vce versa. The followg theorem s prove: Every atural umber except for 0 ad ca be uquely wrtte as a lear combato of cosecutve powers of wth the coeffcets of the lear combato beg - or +. Ths theorem reveals a set of symmetres the teral order of atural umbers whch caot be derved whe studyg atural umbers o the bass of the product. From such a symmetry a method for detfyg large prme umbers s derved. We prove a factorzato test for the odd umbers. Keywords: Number theory Composte umbers Prme umbers. 00 Mathematcs Subect Classfcatos: A4 N05. Itroducto It holds that every product of atural umbers ca also be wrtte as a sum. The verse (.e. each sum of atural umbers ca be wrtte as a product) does ot hold whe s excluded from the product. Ths s due to prme umbers whch ca be wrtte as a product oly the form of p p p. For ths reaso the vestgato of atural umbers should be doe through ther sum ad ot through ther product. Such a vestgato s preseted the preset artcle. We prove that each atural umber ca be wrtte as a sum of three or more cosecutve atural umbers except of the powers of ad the prme umbers. Each power of ad each

2 prme umber caot be wrtte as a sum of three or more cosecutve atural umbers. Prmes play the same role for odd umbers as the powers of for eve umbers ad vce versa. We prove a theorem whch s aalogous to the fudametal theorem of arthmetc whe we study the postve tegers wth respect to addto: Every atural umber wth the excepto of 0 ad ca be wrtte a uque way as a lear combato of cosecutve powers of wth the coeffcets of the lear combato beg - or +. Ths theorem reveals a set of symmetres the teral order of atural umbers whch caot be derved whe studyg atural umbers o the bass of the product. From such a symmetry a method for detfyg large prme umbers s derved. I the last chapter we prove a factorzato test for the odd umbers. I the study of atural umbers f we focus the sum the parameter whch determes the mmum umber of operatos requred for the factorzato of a odd umber s hghlghted. The sequece μ k We cosder the sequece of atural umbers k k k k... k k A For the sequece k k the followg theorem holds: Theorem.. For the sequece k k.. the followg hold:. (.). No elemet of the sequece s a prme umber. 3. No elemet of the sequece s a power of. 4. The rage of the sequece s all atural umbers that are ot prmes ad are ot powers of. k as a sum of atural umbers. Proof... A ad therefore t holds that 3. Also we have that

3 k4 k 3 sce k ad A Thus the product k k s always a product of two atural umbers dfferet tha thus the atural umber k caot be prme. 3. Let that the atural umber k k s a power of. The t exsts k ad equvaletly k such as. (.) Equato (.) ca hold f ad oly f there exst such as k ad equvaletly. (.3) k We elmate from equatos (.3) ad we obta k k whch s mpossble sce the frst part of the equato s a odd umber ad the secod part s a eve umber. Thus the rage of the sequece k 4. We ow prove that the rage of the sequece k does ot clude the powers of. cludes all atural umbers that are ot prmes ad are ot powers of. Let a radom atural umber N whch s ot a prme or a power of. The N ca be wrtte the form 3

4 N where at least oe of the prove that there are always exst k N k. s a odd umber 3. Let be a odd umber ad A such as 3. We wll We cosder the followg two pars of k k k ad : (.4) k k. (.5) For every t holds ether the equalty or the equalty. Thus for each par of aturals where s odd at least oe of the pars k of equatos (.4) (.5) s defed. We ow prove that whe the atural umber s k 0 the the atural umber. For k 0 from equatos (.4) we take ad from equatos (.5) we have that k ad because we obta k 3. k k k of equato (.4) of equato (.5) s k ad addtoally t holds that We ow prove that whe k 0 equatos (.5) the equatos (.4) t s k ad. For k 0 from equatos (.5) we obta 4

5 ad from equatos (.4) we get k. We ow prove that at least oe of the k ad k s postve. Let k 0 k 0. The from equatos (.4) ad (.5) we have that 0 0. (.6) Takg to accout that s odd that s we obta from equaltes (.6) whch s absurd. Thus at least oe of k ad k s postve. For equatos (.4) we take k k k N. For equatos (.5) we obta k k k N. Thus there are always exst k ad A such as N k for every N whch s ot a prme umber ad s ot a power of. Example.. For the atural umber N 40 we have 5

6 N ad from equatos (.4) we get 6 5 k k thus we obta Example.. For the atural umber N 5 N there are two cases. Frst case: N ad from equatos (.4) we obta 34 3 k k 6 3 thus 5 6. Secod case: N ad from equatos (.5) we obta 7 6 k k thus

7 The secod example expresses a geeral property of the sequece k. The more composte a odd umber that s ot prme (or a eve umber that s ot a power of ) s the more are the k Example.3. combatos that geerate t a b c I the trastve property of multplcato whe wrtg a composte odd umber or a eve umber that s ot a power of as a product of two atural umbers we use the same atural umbers :. O the cotrary the atural umber ca be wrtte the form k usg dfferet atural umbers k ad A through equatos (.4) (.5). Ths dfferece betwee the product ad the sum ca also become evdet example.3: From Theorem. the followg corollary s derved: Corollary... Every atural umber whch s ot a power of ad s ot a prme ca be wrtte as the sum of three or more cosecutve atural umbers.. Every power of ad every prme umber caot be wrtte as the sum of three or more cosecutve atural umbers. Proof. Corollary. s a drect cosequece of Theorem.. 7

8 3 The cocept of rearragemet I ths paragraph we preset the cocept of rearragemet of the composte odd umbers ad eve umbers that are ot power of. Moreover we prove some of the cosequeces of the rearragemet the Dophate aalyss. The cocept of rearragemet s gve from the followg defto: Defto 3.. We say that the sequece k k A there exst atural umbers k A k k such as k k s rearraged f. (3.) From equato (.) wrtte the form of k k k k... k two dfferet types of rearragemet are derved: The compresso durg whch decreases wth a smultaeous crease of k. The «decompresso» durg whch creases wth a smultaeous decrease of k. The followg theorem provdes the crtero for the rearragemet. of the sequece k Theorem 3... The sequece k k k k A ca be compressed (3.) f ad oly f there exst whch satsfes the equato k 0. The sequece k k k. (3.3) k A ca be decompressed (3.4) f ad oly f there exst k whch satsfes the equato k 0 k 3. The odd umber s prme f ad oly f the sequece. (3.5) 8

9 k l k A l (3.6) caot be rearraged. 4. The odd s prme f ad oly f the sequece caot be rearraged. (3.7) Proof.. We prove part of the corollary ad smlarly umber ca also be prove. From k equato (4.) we coclude that the sequece exst such as k k. I ths equato the atural umber. Next from equatos (.) we obta k k k k ad after the calculatos we get equato (3.3). belogs to the set ca be compressed f ad oly f there A ad thus 3. The sequece (3.6) s derved from equatos (.4) or (.5) for ad l. Thus the product the oly odd umber s. If the sequece k equato (3.6) caot be rearraged the the odd umber has o dvsors. Thus s prme. Obvously the verse also holds. 4. Frst we prove equatos (3.7). From equato (.) we obta:. I case that the odd umber s prme equatos (.4) (.5) the atural umbers are uque ad from equato (.5) we get k. 9

10 Thus the sequece k caot be rearraged. Coversely f the sequece caot be rearraged the odd umber caot be composte ad thus We ow prove the followg corollary: Corollary 3... The odd umber s prme. odd s decompressed ad compressed f ad oly f the odd umber s composte. (3.8). The eve umber l l odd l 3 l l caot be decompressed whle t compresses f ad oly f the odd umber (3.9) s composte. 3. The eve umber odd l l l l l (3.0) caot be compressed whle t decompresses f ad oly f the odd umber s composte. 4. Every eve umber that s ot a power of ca be wrtte ether the form of equato (3.9) or the form of equato (3.0). 0

11 Proof.. It s derved drectly through umber (4) of Theorem 3.. A secod proof ca be derved through equatos (.4) (.5) sce every composte odd ca be wrtte the form of odds. 3. Let the eve umber l odd. (3.) l From equato (.4) we obta l l k ad sce k k we get (3.) l ad equvaletly l 3. I the secod of equatos (3.) the atural umber obtas the maxmum possble value of ad thus the atural umber k takes the mmum possble value the frst of equatos (3.). Thus the eve umber l caot decompress. If the odd umber s composte the t ca be wrtte the form of odds l. Therefore the atural umber l decompresses sce from equatos (3.) t ca be wrtte the form of k. Smlarly the proof of 3 s derved from equatos (.5). wth 4. From the above proof process t follows that every eve umber that s ot a power of ca be wrtte ether the form of equato (3.9) or the form of equato (3.0). By substtutg P prme equatos of Theorem 3. ad of corollary 3. four sets of equatos are derved each cludg fte mpossble dophate equatos.

12 Example 3.. The odd umber P s prme. Thus combg () of Theorem 3. wth () of corollary 3. we coclude that there s o par wth whch satsfes the dophate equato We ow prove the followg corollary: Corollary 3.. The square of every prme umber ca be uquely wrtte as the sum of cosecutve atural umbers. Proof. For P prme equato (3.5) we obta P P P. (3.3) Accordg wth 4 of Theorem 3. the odd caot be rearraged. Thus the odd ca be uquely wrtte as the sum of cosecutve atural umbers as gve from equato (3.3). Example 3.. The odd P 7 s prme. From equato (3.3) for P 7 we obta ad from equato (.) we get P whch s the oly way whch the odd umber 89 ca be wrtte as a sum of cosecutve atural umbers. 4 Natural umbers as lear combato of cosecutve powers of Accordg to the fudametal theorem of arthmetc every atural umber ca be uquely wrtte as a product of powers of prme umbers. The prevously preseted study reveals a correspodece betwee odd prme umbers ad the powers of. Thus the questo arses whether there exsts a theorem for the powers of correspodg to the fudametal theorem of arthmetc. The aswer s gve by the followg theorem: Theorem 4.. Every atural umber wth the excepto of 0 ad ca be uquely wrtte as a lear combato of cosecutve powers of wth the coeffcets of the lear combato beg - or +. Proof. Let the odd umber as gve from equato

13 From equato (4.) for we obta. (4.) We ow exame the case where (4.) ca obta s. The lowest value that the odd umber of equato m... m. (4.) The largest value that the odd umber of equato (4.) ca obta s max... max. (4.3) Thus for the odd umbers m of equato (4.) the followg equalty holds. (4.4) The umber N N N max of odd umbers the closed terval max m s. (4.5) The tegers 0... equato (4.) ca take oly two values thus equato (4.) gves exactly N equato (4.) gves all odd umbers the terval odd umbers. Therefore for every. We ow prove the theorem for the eve umbers. Every eve umber whch s a power of ca be uquely wrtte the form of. We ow cosder the case where the eve umber s ot a power of. I that case accordg to corollary 3. the eve umber s wrtte the form of odd l l. (4.6) 3

14 We ow prove that the eve umber ca be uquely wrtte the form of equato (4.6). If we assume that the eve umber ca be wrtte the form of ' ' l l l l ' ' ' ' l l ' ll ( ) odd (4.7) the we obta ' ll ' l l ' ' whch s mpossble sce the frst part of ths equato s eve ad the secod odd. Thus t s ad we take that from equato (4.7). Therefore every eve umber that s ot a power of ca be uquely wrtte the form of equato (4.6). The odd umber of equato (4.6) ca be uquely wrtte the form of equato (4.) thus from equato (4.6) t s derved that every eve umber that s ot a power of ca be uquely wrtte the form of equato l l ' ' l l l (4.8) ad equvaletly l l l 0 l l (4.9) 0 For we take 0 thus t ca be wrtte two ways the form of equato (4.). Both the odds of equato (4.) ad the eves of the equato (4.8) are postve. Thus 0 caot be wrtte ether the form of equato (4.) or the form of equato (4.8). I order to wrte a odd umber 3 the form of equato (4.) we tally defe the from equalty (4.4). The we calculate the sum 4

15 . If t holds that we add the whereas f t holds that the we subtract t. By repeatg the process exactly tmes we wrte the odd umber the form of equato (4.). The umber of steps eeded order to wrte the odd umber the form of equato (4.) s extremely low compared to the magtude of the odd umber as derved from equalty (4.4). Example 4.. For the odd umber 3 4 thus 3. The we have (thus (thus (thus form F s Fermat umbers m F s 3 s added) we obta from equalty (4.4) s subtracted) 0 s added) ca be wrtte drectly the form of equato (4.) sce they are of the s s s s s s 3 m.... (4.0) s Mersee umbers form M p max M p ca be wrtte drectly the form of equato (4.) sce they are of the 3 p max.... (4.) p prme I order to wrte a eve umber that s ot a power of the form of equato (4.) tally t s cosecutvely dvded by ad t takes of the form of equato (4.6). The we wrte the odd umber the form of equato (4.). Example 4.. By cosecutvely dvdg the eve umber 368 by we obta The we wrte the odd umber 3 the form of equato (4.)

16 ad we get Ths equato gves the uque way whch the eve umber form of equato (4.9). From equalty (4.4) we obta log log log from whch we get log log log log ad fally 368 ca be wrtte the log log (4.) log where log the teger part of log. log We ow gve the followg defto: Defto 4.. We defe as the cougate of the odd (4.3) the odd 0... for whch t holds 0 (4.4) 6

17 0.... (4.5) k k k For cougate odds the followg corollary holds: Corollary 4.. For the cougate odds ad the followg hold:.. (4.6). 3. (4.7) 3. s dvsble by 3 f ad oly f s dvsble by 3. Proof.. The of the corollary s a mmedate cosequece of defto 4... From equatos (4.3) (4.4) ad (4.5) we get ad equvaletly If the odd s dvsble by 3the t s wrtte the form 3 x x odd ad from equato (4.7) we get 3x 3 verse. ad equvaletly 3 x. Smlarly we ca prove the 5 The T symmetry ad a method for defg large prme umbers We ow gve the followg defto: Defto 5.. Defe as symmetry every specfc algorthm whch determes the sgs of 0... equato (4.): Next we develop a specfc symmetry the T symmetry. If the atural umber the equato (4.) s ot a prme ad s ot a power of the equato (.) gves k k k k k... k. (5.) k A

18 We defe the odd umber T k T k as follows: I the rght sde of equato (4.) from left to rght we take k sgs - ad the (k+) sgs + (k+) sgs - (k+3) sgs + etc. accordg to the rght sde of equato (5.). After makg some calculatos we have T k T k 0 k A k k 0 k k k 0 k k k 0 (5.) ad T T k 3 T k k ad equvaletly k k k 0 0 T T k k k k 0 k k k 0 k A. (5.3) We wrte the equato (5.) the form k A k k k k k... k. (5.4) We defe the odd umber T k T k T k T k by the same way as we defed but the sgs equato (4.) are ow determed accordg to the rght sde of equato (5.4) (k+) sgs - (k+-) sgs + (k+-) sgs - (k+-3) sgs + etc. After makg some calculatos we have 8

19 k T T k 0 k A k k 0 k k k 0 k k k 0 (5.5) ad T T k 3 T k k ad equvaletly k k k 0 0 T T k k k k 0 k k k 0 k A. (5.6) Equatos (5.) (5.3) (5.5) ad (5.6) defe the T symmetry. A method for the determato of large prme umbers emerges from the study we preseted. Ths method s completely dfferet from prevous methods [-5]. For the T symmetry holds: k There are pars A 3 4 L L (5.7) for whch oe or more of T k T k T k T We wll preset three examples:. The umber k are prme umbers. T 5 = s a prme. The umber T = s a prme

20 . The umber T 34 = s a prme. 3. The umber T = (74 dgts) s a prme. The umber D of dgts of the prmes calculated by the method s of order k D Dk k log log. (5.8) T =49. Also t The smallest prme umber gve by the method s does't gve prme umbers Fermat ad Mersee. We ow cte some remarkable propertes of the T symmetry. Whe the umbers of the T symmetry are ot prmes wth hgh probablty oe or more of them are the product of a set of small prmes wth a large prme (wth rato of the umber of dgts at least 3: the decmal system). We gve a example for =4 ad k= Example 5.. T 4 = T 4 = T 4 = T = T 34 = (9 dgts) s a prme T = (9 dgts) s a prme. T 44 = T 54 = ( dgts) s a prme T = T 64 = T 74 = T =

21 T 84 = T 4 9. = T 4 0. = T 4 = T 4 T 3 4. T 34 T 4 4. = = = = T = T 64 = T 74 = T 84 = T 94 = T 04 = T 4 = (33 dgts). T 4 = (3 dgts) 0. 4 T = (33 dgts). T 34 = (38 dgts) s a prme. 34 T = (34 dgts) 34 T = (36 dgts). For 3 4 L L (5.9) the umbers of the T symmetry have 3 as a factor. I these cases we factorze the umbers of the T symmetry order to detfy the oes whch are the product of a set of small prmes wth a

22 large prme (wth rato of the umber of dgts at least 3: the decmal system). We gve a example for L=0 ad k= Example 5... T 3 T 3. =3 37. =3 53. T = T 53 = T T T = = = T 3 = T 3 = T 33 = T 43 = T 63 = T 73 = T 83 = T 93 = T = T = T 03 = T 3 = T = T 5 =

23 T 33 = T = T 53. = T 63 = T 63 T T = = = T 83 = T = T 33 = T 33 = (39 dgts) T 333 = (37 dgts) N 333 T = (33 dgts). Fermat ad Mersee for odds N 3 of the form N ad respectvely chose the values of for whch the odd N s does ot have 3 as a factor ( s ad prme respectvely). Ths has as a frstly cosequece that the Fermat ad Mersee umbers are ot dvsble by 3 that s they are ot dvsble by ⅓ of the odd umbers (that are smaller tha N ). Ths o-dvsblty by 3 s a property of the umbers of the T symmetry for =5. Cosequetly the odds 5 T k 5 T k 5 T k T k T k 5 5 T k T k 5 k are ot dvsble by ⅓ of the odd umbers (that are smaller tha 5 T k 5 k ). Because of ths the method s partcularly effcet for =5. We gve a example for =5 ad for small values of k k= Example T 5 = (9 dgts) s a prme.. T 35 = (dgts) s a prme 3

24 35 T = ( dgts) s a prme. T = (3 dgts) s a prme. T = (0 dgts) s a prme. T 95 = ( dgts) s a prme. 5. T = (3 dgts) s a prme. T 5 7. = (5 dgts) s a prme T 5 T 5 8. = (5 dgts) s a prme. = (7 dgts) s a prme. T = (34 dgts) s a prme. T 85 = (38 dgts) s a prme. 0. From the detty of the Eucldea dvso we have that the equatos 3 4 L L 4 L L 4 L L 4 L L gve all values for A For 4 L L the umbers of T symmetry gve prme umbers oly for eve values of k: k S 4L S L. (5.0) For 4 L L the umbers of T symmetry gve prme umbers for both eve ad odd values of k: k S4L SL. (5.) For 4 L L the umbers of T symmetry gve prme umbers oly for odd values of k: k S 4 L SL. (5.) 4

25 The values of sequece μ(k ) for the pars (k ) of equatos (5.0) (5.) ad (5.) are odd umbers. So the umbers of T symmetry gve prme umbers oly cases where the sequece k k I equatos (5.) (5.3) (5.5) ad (5.6) s a odd umber. From the above study t emerges that the method s appled two ways: a. We factorze the umbers of the T symmetry ad detfy the oes that are products of a set of prme umbers wth a comparatvely larger prme umber. b. We detfy the prme umbers of the T symmetry va a prmalty test whe the equatos (5.0) (5.) (5.) hold. We suggest both cases that a specfc A should be chose ad the the values k= 3 ca be gve equatos (5.) (5.3) (5.5) ad (5.6). The method may be k further vestgated for the form of the pars (5.6). A equatos (5.) (5.3) (5.5) ad The observatos above have hgh theoretcal terest but they have ot bee completely proved. Durg the applcato of the method t s ecessary a prmalty test to be doe for all possble prmes of T symmetry. 6. A factorzato test for the composte odd umbers The corollary gves a factorzato test for the odd umbers 9. Corollary 6... Every composte odd umber 9ca be wrtte the form hc h c A c h c (6.) f ad oly f there exsts a odd umber f such that h 8 f f f odd 8 f 8 h ad the f h c. (6.) (6.3) 5

26 . The bggest umber S=S(Π) of operatos requred for the factorzato of the odd umber Π depeds o the value of the parameter h c A ad derves from the equato hc 8 8 h S S h c h. (6.4) Proof.. Accordg to theorem. every composte odd umber ca be wrtte the form of the equato (6.) ad we have c h c h 0. (6.5) Ths equato s of secod order wth respect to c A s a square of a atural umber: ad the determat D of the equato (6.5) 8 D h f. (6.6) From the equatos (6.5) ad (6.6) we have f h c A whch s the equato (6.3). I equato (6.3) the atural umber h s odd ad cosequetly f s also odd. From the equato (6.6) we have that 8 f ad fally 8 f. From the equato (6.) we have c h c ad so c 8 3 ad combato wth the equato (6.3) we have f h 8 3 Ad fally we have f 8 h.. From the equalty of relato (6.) we have that the odd umber f belogs to the closed terval Δ=Δ(Π): f h c 8 8 h. (6.7) 6

27 Cosequetly the bggest umber of operato requred s 8 8 h S S h c h the case where the umber f takes all odd values the terval Δ. The factorzato of the odd umber Π ca be doe by two ways: By gvg to the atural umber h the values h= 3 the equato (6.) utl we have a odd value for f. The from equato (6.3) we calculate c A ad take the odd Π factorzed the form hc c h c ( factors). By gvg to the odd umber f values f 8 utl we have a atural umber equato (6.). The from the equato (6.3) we calculate c A ad take the odd Π factorzed c h c the form hc ( factors). From the equato (6.4) we coclude that the bggest umber of operatos requred S=S(Π) for the factorzato of the odd umber Π s mmzed whe the umber h takes the smallest possble value the rearragemets of Π=μ(h c) (see chapter 3). By comparg our factorzato test wth the seve of Eratosthees: the umber of operatos requred for the h from factorzato of a odd umber Π by the seve of Eratosthees s approxmately l. So the factorzato test s effcet for the odd umbers Π for whch we have S h c ad equvaletly l h. (6.8) l As we ca coclude from equato (6.4). The test s very effectve for the odd umbers Π for whch we have h. (6.9) l From equatos (.4) ad (.5) where k=h ad =c we have that parameter h takes small values ad equvaletly the test s effectve whe a odd umber Π s a product of two odd umbers χ ad ψ χ<ψ ad ψ be about twce as hgh as χ. Thw observato leads to the cocept of rearragemet multpler : If the odd umber Π caot be factorzed by the test the we 7

28 multply t by a odd ξ (rearragemet multpler) so that ξπ s product of two odd umbers χ ad ψ χ<ψ ad ψ be about twce as hgh as χ. The we factorze ξπ by the test. I factorzato of ξπ the bggest factor of Π s appeared. Next we ca see four examples. Example 6.. We apply the test wth the frst way metoed above for the odd Π= The test factorzes 6 operatos: = Example 6.. We apply the test wth the frst way metoed above for the odd Π= The test factorzes operatos: Π= = We apply the test wth the secod way metoed above for the odd Π= The test factorzes 34 operatos: Π= = Example 6.3. We apply the test wth the frst way metoed above for the odd Π=97. The test factorzes 334 operatos whch s a extremely hgh umber for such a small umber. We apply the test wth the secod way metoed above for the odd Π=97. The test factorzes 4 operatos. For ξ=83 we ca foctorze 83 Π=83 97=989 ad by applyg the test by the frst way metoed above we ca take ust 5 operatos performed 989=70 4. The odd umber 70 s the bggest factor of the odd Π=97. Example 6.4. We apply the test wth the frst way metoed above for the odd Π= The test factorzes 8504 operatos whch s a extremely hgh umber for such a small umber. We apply the test wth the secod way metoed above for the odd Π= The test factorzes 5769 operatos. For ξ=77 we ca foctorze 77 Π= = ad by applyg the test by the frst way metoed above we ca take 0 operatos performed = The odd umber s the bggest factor of the odd Π= For ξ=79 we ca foctorze 79 Π= = ad by applyg the test by the frst way metoed above we ca take ust operato performed = The odd umber s the bggest factor of the odd Π= For ξ=7 we ca foctorze 7 Π= = ad by applyg the test by the frst way metoed above we ca take 0 operato performed = The odd umber s the bggest factor of the odd Π= I order to fd the rearragemet multpler we may mplemet some dfferet methods. I ths paper we wll ot make meto to these methods. 8

29 The results we have set out as well as the applcatos of Chapters 5 ad 6 ca be further explored. Ths s expected because ths s the frst tme we study the atural umbers by ther sum ad ot by ther product. Refereces [] Apostol Tom M. Itroducto to aalytc umber theory. Sprger Scece & Busess Meda 03. [] Cradall Rchard ad Carl B. Pomerace. Prme umbers: a computatoal perspectve. Vol. 8. Sprger Scece & Busess Meda 006. [3] Gurevch Alexader ad Bors Kuyavskĭ. "Prmalty testg through algebrac groups." Arch. der Math (009): 555. [4] Rempe-Glle Lasse ad Rebecca Waldecker. Prmalty testg for begers. Amer. Math. Soc. 04. [5] Schoof Ree. Four prmalty testg algorthms Algorthmc Number Theory: Lattces Number Felds Curves ad Cryptography