A study of the sum of three or more consecutive natural numbers

Transcription

1 A study of the sum of three or more cosecutve atural umbers 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. he 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. he 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 +. hs 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 atural umbers. Keywords: Number theory Composte umbers Prme umbers. 00 Mathematcs Subject Classfcatos: A4 N05. Itroducto It holds that every product of atural umbers ca also be wrtte as a sum. he verse (.e. each sum of atural umbers ca be wrtte as a product) does ot hold whe s excluded from the product. hs s due to prme umbers p whch ca be wrtte as a product oly the form of 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. he sum of cosecutve atural umbers has bee studed the past [ ]. I ths paper we studyg the sum of 3 or more cosecutve atural umbers.

2 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 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 +. hs 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. he sequece μk We cosder the sequece of atural umbers k k k k... k k A For the sequece k k the followg theorem holds: heorem.. 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. he 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.

3 Also we have that k4 k 3 sce k k ad A hus the product k s always a product of two atural umbers dfferet tha thus the atural umber caot be prme. 3. Let that the atural umber k k s a power of. he t exsts such as k ad equvaletly k k. (.) 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. hus the rage of the sequece k does ot clude the powers of. 3

4 4. We ow prove that the rage of the sequece k 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. he N ca be wrtte the form 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. hus 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 4

5 We ow prove that whe. For k 0 k 0 from equatos (.5) we obta equatos (.5) the equatos (.4) t s k ad 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. he from equatos (.4) ad (.5) we have that 0 0. (.6) akg to accout that s odd that s we obta from equaltes (.6) whch s absurd. hus 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. hus there are always exst k ad A such as 5

6 N k for every N whch s ot a prme umber ad s ot a power of. Example.. For the atural umber N ad from equatos (.4) we get 6 5 k k thus we obta N 40 we have Example.. For the atural umber N there are two cases. Frst case: N N 5 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

7 thus he secod example expresses a geeral property of the sequece k. he 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). hs dfferece betwee the product ad the sum ca also become evdet example.3: From heorem. the followg corollares are 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 heorem.. 7

8 Corollary.. he sequece k ca be wrtte as a dfferece of two tragle umbers. Proof. From equato (.) we obta k k k k k k k k A (.7) 3 he 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. he 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: he compresso durg whch decreases wth a smultaeous crease of k. he «decompresso» durg whch creases wth a smultaeous decrease of k. he followg theorem provdes the crtero for the rearragemet. of the sequece k heorem 3... he sequece k k k k A ca be compressed (3.) f ad oly f there exst whch satsfes the equato k 0. he sequece k k k. (3.3) k A ca be decompressed (3.4) 8

9 f ad oly f there exst k whch satsfes the equato k 0 k. (3.5) 3. he odd umber k l k A l s prme f ad oly f the sequece (3.6) caot be rearraged. 4. he 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 equato (4.) we coclude that the sequece k 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). ca be compressed f ad oly f there belogs to the set A ad thus 3. he sequece (3.6) s derved from equatos (.4) or (.5) for ad l. hus the product the oly odd umber s. If the sequece k equato (3.6) caot be rearraged the the odd umber has o dvsors. hus s prme. Obvously the verse also holds. 4. Frst we prove equatos (3.7). From equato (.) we obta: 9

10 . I case that the odd umber s prme equatos (.4) (.5) the atural umbers are uque ad from equato (.5) we get k. hus 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... he odd umber s prme. odd s decompressed ad compressed f ad oly f the odd umber s composte. (3.8). he 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. he eve umber 0

11 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). Proof.. It s derved drectly through umber (4) of heorem 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.). hus the eve umber l caot decompress. If the odd umber s composte the t ca be wrtte the form of odds l. herefore the atural umber l

12 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 heorem 3. ad of corollary 3. four sets of equatos are derved each cludg fte mpossble dophate equatos. Example 3.. he odd umber P s prme. hus combg () of heorem 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.. he 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 heorem 3. the odd caot be rearraged. hus the odd ca be uquely wrtte as the sum of cosecutve atural umbers as gve from equato (3.3). Example 3.. he 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. he prevously preseted study reveals a correspodece betwee odd prme umbers ad the powers of. hus the questo arses whether there exsts a theorem for the powers of correspodg to the fudametal theorem of arthmetc. he aswer s gve by the followg theorem:

13 heorem 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 From equato (4.) for we obta. (4.) We ow exame the case where (4.) ca obta s. he lowest value that the odd umber of equato m... m. (4.) he largest value that the odd umber of equato (4.) ca obta s max... max. (4.3) hus for the odd umbers m of equato (4.) the followg equalty holds. (4.4) he umber N N N max of odd umbers the closed terval max m s. (4.5) he 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. herefore 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 3

14 eve umber s ot a power of. I that case accordg to corollary 3. the eve umber s wrtte the form of l odd l. (4.6) 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. hus t s ' ad we take that from equato (4.7). herefore every eve umber that s ot a power of ca be uquely wrtte the form of equato (4.6). he 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 4

15 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. hus 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). he we calculate the sum. 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.). he 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. he we have (thus (thus (thus form 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 F s.... (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). he we wrte the odd umber the form of equato (4.). 5

16 Example 4.. By cosecutvely dvdg the eve umber 368 by we obta he we wrte the odd umber 3 the form of equato (4.) ad we get hs 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.) where log log the teger part of log. log We ow gve the followg defto: Defto 4.. We defe as the cojugate of the odd (4.3) the odd 6

17 j j 0... for whch t holds j0 j j (4.4) (4.5) k k k For cojugate odds the followg corollary holds: Corollary 4.. For the cojugate odds ad. the followg hold:. (4.6). 3. (4.7) 3. s dvsble by 3 f ad oly f s dvsble by 3. Proof.. he 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 he 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 symmetry.. 7

18 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 We defe the odd umber k 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 k k j0 k A j k j k 0 j k k k j j0 j k k k j j0 (5.) ad k 3 k k ad equvaletly k j k j k 0 j0 k j k k k j j0 j k k k j j0 k A. (5.3) We wrte the equato (5.) the form 8

19 k A k k k k k... k. (5.4) We defe the odd umber k k k 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 k k j0 k A j k j k 0 j k k k j j j0 j k k k j j j0 (5.5) ad k 3 k k ad equvaletly k j k j k 0 j0 k j k k k j j j0 j k k k j j j0 k A. (5.6) Equatos (5.) (5.3) (5.5) ad (5.6) defe the symmetry. A method for the determato of large prme umbers emerges from the study we preseted. hs method s completely dfferet from prevous methods [3-7]. For the symmetry holds: k here are pars A 3 4 L L (5.7) 9

20 for whch oe or more of k k k k are prme umbers. We wll preset three examples:. he umber = s a prme. he umber = s a prme he umber = s a prme. 3. he umber = (74 dgts) s a prme. he umber D of dgts of the prmes calculated by the method s of order k D Dk k log log. (5.8) =49. Also t he smallest prme umber gve by the method s does't gve prme umbers Fermat ad Mersee. We ow cte some remarkable propertes of the symmetry. Whe the umbers of the 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 = = = = = (9 dgts) s a prme = (9 dgts) s a prme. 0

21 44 4. = = ( dgts) s a prme = = = = = = = = = = = = = = = = = = = (33 dgts). 4 = (3 dgts) 0. 4 = (33 dgts).

22 34. = (38 dgts) s a prme For = (34 dgts) = (36 dgts). 3 4 L L (5.9) the umbers of the symmetry have 3 as a factor. I these cases we factorze the umbers of the symmetry order to detfy the oes whch 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 L=0 ad k= Example = = = = = = = = = = = = = = =

23 = = = = = = = = = = = = = = = = = (39 dgts) = (37 dgts) N 333 = (33 dgts). Fermat ad Mersee for odds N 3 of the form N ad respectvely chose the values of for whch the odd N frstly s does ot have 3 as a factor ( s ad prme respectvely). hs has as a 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 ). hs o-dvsblty by 3 s a property of the umbers of the symmetry for =5. Cosequetly the odds 5 k k 5 3

24 k 5 k5 k 5 k 5 k are ot dvsble by ⅓ of the odd umbers (that are smaller tha k5 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 = (9 dgts) s a prme.. 35 = (dgts) s a prme = ( dgts) s a prme. = (3 dgts) s a prme = (0 dgts) s a prme = ( dgts) s a prme. 05 = (3 dgts) s a prme. 5 = (5 dgts) s a prme 7. 5 = (5 dgts) s a prme. 5 = (7 dgts) s a prme = (34 dgts) s a prme = (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 symmetry gve prme umbers oly for eve values of k: k S 4L S L. (5.0) 4

25 For values of k: 4 L L the umbers of symmetry gve prme umbers for both eve ad odd k S4L SL For. (5.) 4 L L the umbers of symmetry gve prme umbers oly for odd values of k: k S 4 L SL. (5.) he values of sequece μ(k) for the pars (k) of equatos (5.0) (5.) ad (5.) are odd umbers. So the umbers of 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 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 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). he method may be k further vestgated for the form of the pars (5.6). A equatos (5.) (5.3) (5.5) ad he 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 symmetry. 6. A factorzato test he corollary gves a factorzato test for the odd umbers 9. Corollary 6... Every odd umber 9 s composte f ad oly f there exsts a odd umber such that f 5

26 h 8 f h f f odd 8 f 8 h (6.) ad the c h c hc (6.) f h c A. (6.3). he bggest umber S=S(Π) of operatos requred for the factorzato of the odd umber Π depeds o the value of the parameter h ad derves from the equato 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 hc h c ad we have A c h c h c h c 0. (6.5) hs equato s of secod order wth respect to c A ad the determat D of the equato (6.5) s a square of a atural umber: 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 6

27 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) 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. he 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. he from equato (6.3) we calculate c A ad take the odd Π factorzed c h c the form hc ( factors). By gvg to the odd umber f values f 8 utl we have a atural umber h from equato (6.). he 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 factorzato of a odd umber Π by the seve of Eratosthees s approxmately l. herefor the factorzato test s effcet for the odd umbers Π for whch we have S h c l 7

28 ad equvaletly h l. (6.8) As we ca coclude from equato (6.4). he test s very effectve for the odd umbers Π for whch we have h l. (6.9) 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 Π=χψ χ<ψ ad ψ be about twce as hgh as χ:. (6.0) hs observato leads to the cocept of rearragemet multpler : If the odd umber Π caot be factorzed by the test the we 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. he we factorze ξπ by the test. I factorzato of ξπ the bggest factor of Π s appeared. Next we ca see seve examples. Example 6.. We apply the test wth the frst way metoed above for the odd Π= he test factorzes just operato: = Example 6.. We apply the test wth the frst way metoed above for the odd Π= he test factorzes 6 operatos: = Example 6.3. We apply the test wth the frst way metoed above for the odd Π= he test factorzes operatos: Π= = We apply the test wth the secod way metoed above for the odd Π= he test factorzes 34 operatos: Π= = Example 6.4. We apply the test wth the frst way metoed above for the odd Π= (56 dgts). he test factorzes 5 operatos: Π=

29 Example 6.5. We apply the test wth the frst way metoed above for the odd Π= (30dgts). he test factorzes 5 operatos: Π= Example 6.6. We apply the test wth the frst way metoed above for the odd Π=97. he 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. he test factorzes 4 operatos. For ξ= we ca factorzes Π= 97= 5057 ad by applyg the test by the frst way metoed above we ca take 7 operatos performed ξπ= 5057 = he odd umber 70 s the bggest factor of the odd Π=97. For ξ=83 we ca factorzes 83 Π=83 97=989 ad by applyg the test by the frst way metoed above we ca take 5 operatos performed ξπ= 989=70 4. he odd umber 70 s the bggest factor of the odd Π=97. Example 6.7. We apply the test wth the frst way metoed above for the odd Π= he 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= he test factorzes 5769 operatos. For ξ=77 we ca factorzes 77 Π= = ad by applyg the test by the frst way metoed above we ca take 0 operatos performed ξπ= = he odd umber s the bggest factor of the odd Π= For ξ=79 we ca factorzes 79 Π= = ad by applyg the test by the frst way metoed above we ca take just operato performed ξπ= = he odd umber s the bggest factor of the odd Π= For ξ=7 we ca factorzes 7 Π= = ad by applyg the test by the frst way metoed above we ca take 0 operatos performed ξπ= 9

30 = he odd umber s the bggest factor of the odd Π= For ξ=79 we ca factorzes 79 Π= = ad by applyg the test by the frst way metoed above we ca take 766 operatos performed ξπ= = he odd umber s the bggest factor of the odd Π= We prove ow the followg corollary for the rearragemet multpler: Corollary 6.. For every odd umber Π Π=χψ χ<ψ χ ψ odd umbers the rearragemet multplers ξ are always pars for whch the followg equatos hold 4 odd (6.) 4 odd umbers. (6.) Prof. For the odd umber Π Π=χψ χ<ψ χ ψ odd umbers we have ξπ=ξχψ. (6.3) From equato (6.4) for the odd umber ξπ ad from equatos (.4) (.5) we have that the odd umber ξπ s factorzed by the test the followg cases: For ad we have. (6.4) For ad we have. (6.5) By cacellg χ ad ψ from the equatos (6.4) (6.5) we have 4. (6.6) I relato (6.6) the umbers ad are odd ad cosequetly we have 4 whch s exactly the equato (6.). he equato (6.) results from relato (6.) whereas the umbers ad are odd. 30

31 For the odd umbers of the examples ad 6.5 we have that the relato (6.0) holds. I examples 6.6 ad 6.7 we ca see the cosequeces of the corollary 6.. We prove ow the followg corollary: Corollary 6.3. Every odd umber N ad h so that to have hh N N 9 s composte f ad oly f there exst atural umbers 8 8 N h. (6.7) hn Proof. We set f N N equato (6.) ad by makg calculatos we take the equato (6.7). (6.8) he equatos (6.) ad (6.7) are equvalet. From the equato (6.7) ad by our test we have that the odd umber Π ca be wrtte as a dfferece of two tragle umbers. Cosequetly the test ca factorze odd umbers whch caot be factorzed by tests that are based o dffereces ad o sums of atural umber squares. We uderle that our test gves the odd umber Π as a product of two factors so may as the factors of the sequece μ(hc) are. he test ca also appled to eve umbers. As the case of equatos (6.) our test ca be appled two ways: For the atural umber h we gve values h=3 equato (6.7) utl a atural umber N to fd out 8 8 N h. Next from the equato (6.8) we calculate the odd umber f ad from equato (6.3) we calculate c A ad take the odd umber Π factorzed the form Π=μ(hc). 8 We gve N values N equato (6.7) utl a atural umber h to fd out. Next from equato (6.8) ext we calculate the odd umber f ad from equato (6.3) we calculate c A ad take the odd umber Π factorzed the form Π=μ(hc). From corollares 6. ad 6.3 we ca fd out the followg results for the usage of the test: 3

32 . he test factorzes odd umbers Π for whch the equato (6.0) holds. he umber of calculatos requred for the factorzato depeds o the teral structure of the odd umber Π (from the quotet ψ/χ) ad t s depedet from the sze of the odd umber Π.. he test ca factorze odd umbers for whch the equato (6.0) does t hold the followg way: From equato (6.) we have that for every odd umber caddate multpler whch s rejected mmedately all odd caddate multplers of the form 4 are rejected. Cosequetly every tral we do for a caddate multpler s rejected the the caddate multplers 35 testg as caddate multpler the ext odd umber whch s s equvalet to three trals. If are rejected too. So we go o by 7. If 7 s rejected the mmedately the caddate multplers 7 9 are rejected too. he we ca test as caddate multpler 9 from ad so o. he tal value of depeded o the value of the odd umber Π. I the same way we have a descedg sequece of caddate whch are rejected by relato (6.). 3. Usg a rearragemet multpler the test ca factorze odd umbers Q whch caot be factorzed by tests that are based o dffereces ad o sums of atural umber squares. he tal values of ad are depeded o the value of odd Q. he above preseted factorzato test may be combed wth factorzato tests that are based o dffereces ad o sums of atural umbers squares. hs combato could solve the problem of atural umbers factorzato. For example classcal tests experece a dffculty factorzg the odd umber Π (30 dgts) of example 6.5 ( cotrary they ca easly factorze the odd umbers 8. 0 operatos). O the = ad = hus by combg the two tests we obta the factorzato of the odd umber Π: Π=

33 (30dgts) = ( ) ( ). he tests that are based o dffereces ad o sums of atural umber squares are hghly effcet whe the followg relatoshps hold (6.9) or (6.0) for the odd umbers Π=χψ χ<ψ χ ψ odd umbers. herefore there s a emergg eed of studyg the assocato betwee the two tests based o relatoshps (6.0) ad (6.9) (6.0). hs study ca provde the tal values of rearragemet multplers ad for the odd umbers Q that caot be factorzed by the classcal tests. I some frst applcatos that have bee coducted the determato of the tal value of was based o the statstcal estmato of the relatoshp betwee ad χ equato (6.4) for Π=Q Q. (6.) he results we have set out as well as the applcatos of Chapters 5 ad 6 ca be further explored. hs s expected because ths s the frst tme we study the atural umbers by ther sum ad ot by ther product. Refereces [] LeVeque W. J. "O represetatos as a sum of cosecutve tegers." Caad. J. Math (950): [] Parshall Kare Huger. "Amerca's frst school of mathematcal research: James Joseph Sylvester at he Johs Hopks Uversty " Archve for Hstory of Exact Sceces 38. (988):

34 [3] Apostol om M. Itroducto to aalytc umber theory. Sprger Scece & Busess Meda 03. [4] Cradall Rchard ad Carl B. Pomerace. Prme umbers: a computatoal perspectve. Vol. 8. Sprger Scece & Busess Meda 006. [5] Gurevch Alexader ad Bors Kuyavskĭ. "Prmalty testg through algebrac groups." Arch. der Math (009): 555. [6] Rempe-Glle Lasse ad Rebecca Waldecker. Prmalty testg for begers. Amer. Math. Soc. 04. [7] Schoof Ree. Four prmalty testg algorthms Algorthmc Number heory: Lattces Number Felds Curves ad Cryptography