Fermat s Theorem and its relation to the Binomial Theorem

Transcription

1 Fermat s Theorem ad its relatio to the Biomial Theorem Charles Keyser, 9/9/16 Update 9//16 Space-Time Forum BuleriaChk@aol.com Fermat s Last Theorem The equatio The Biomial Theorem c a b caot be true for a, b, c ad positive itegers ad > The purpose of this paper is show the relatio of Fermat s theorem to the coservatio of particle coutig i the cotext of particle physics. To this ed, I first relate the Special Theory of Relativity to the two particle vector space of positive particles (a,b) modeled as positive itegers, ad show it relates to particle coutig. I the show that the space is covered by either Pythagorea triples i the case where the particle cout is coserved or the Biomial Theorem for the case where the particles are ot. I the show how the expasio is modeled for the case >, with the Biomial Theorem icorporatig o-liear elemets due to the outer products that model the particle iteractios, with the result that the iteractio term is ever zero, thus provig Fermat s theorem Particles ad the Special Theory of Relativity For STR, a particle "exists" oly if it is positive defiite (squared); i.e., greater tha zero. It ca oly be "destroyed" by i = -1. That s because a existig particle is a coserved "eergy". To see this, assume m is a positive iteger (particle) The first order relatio for classical mometum is P = mv, where the velocity ca be either positive or egative. If it is zero, mometum is zero, but it says othig about the mass of the particle. Eistei assumed the existece of the particle as a positive elemet, m = ct where c is the mass creatio rate, ad t is the time it takes to create the mass (the questio about "who" is a distractio). For two particles, cosider each with a positive ad egative mometum ( v ) respectively. If the particles simply pass each other without iteractio, othig chages physically

2 (although the labels ca be chaged, the coordiates are irrelevat, sice the velocity has already bee assiged) However, if they collide ito a sigle statioary mass, the eergy thus created is give by E = mv with a "kietic" eergy assiged to each particle before the collisio of E = (1/)mv For Eistei ad Special Relativity (via the photoelectric effect), the Mometum ad Eergy of a sigle photo- equivalet mass is give by P mc ad E m c h. (Note: This is a secod order relatio i eergy. I Newto's law, E = 1/ mv, ad the classical (ad relativistic) mass is assumed to be positive. Dirac itroduced the possibility of egative mass, with the total relativistic eergy the ecapsulated i the Dirac Gamma matrices (together with spi).) The eergy per uit rest mass of a sigle particle i STR is the give by c E / m is a positive defiite ivariat quatity, ad ca be modeled mathematically by a positive iteger so that particles are give by c E / m. This To show the relatio betwee two such positive particles, two variables are ecessary, a ad b, both positive itegers, such that / ad a E m b E / m Each of the particles (a,b) ca be thought of as idepedet sets (umber lies) of positive particles {a} ad {b} These quatities ca be added together, so that S ab a b of particles is coserved i the Pythagorea Triple (a,b,c), where c a b c a b Settig a = 1 ad b = 1 yields = (positive itegers)..., ad the total umber ; i.e., Note: If ay of the followig substitutios are NOT (positive) itegers, the Fermat's Theorem is true by default, sice oly positive itegers are allowed. Not oly that, particle cout would ot be coserved. If such a positive iteger does ot exist, the the Biomial Theorem is false. If the umber of particles is differet for a ad b, the sum ca be represeted by m c ma b with the scalig factor {m}, a positive iteger/set. (Biary products of these sets rage over all elemets),

3 The (retaiig idetity of particles), the c ( ) m ma b ma mab b c m a b, so that Here m chages the value of the iteger (set) {a}, ad the expasio icludes the m iteractio term mab, so the relatio is ot a Pythagorea Triple. ma ca be replaced by a iteger, a' ma, so that c ( a b) The terms ca the simply be relabeled agai so that c' c, a' a (i.e., without chagig the values the labels represet), so that c a b a ab b for the case where the form is NOT a Pythagorea triple (ad is addressed by the Biomial Theorem which icludes the iteractio term ab ; for the Pythagorea triples, the iteractio term vaishes. This ca be represeted i a two-dimesioal iteger space by the orthogoal uit vectors i ad j. The relatio betwee vectors i this space is give by the relatio: c ai b j, where c is the resultat of the vector sum ai b j. Because all elemets are positive itegers, these vectors ca be show o a twodimesioal plae, with the positive iteger set a o the positive horizotal axis ad the positive iteger set b o the vertical axis: Pythagorea Triple: I this case, the vectors form a right triagle with the legs parallel to the uit vectors so that c ( ai b j) ( ai b j) a ( i i) ab( i j) b ( j j) a b, ad the iteractio term ab( i j) vaishes (i.e., there is o iteractio betwee the itegers as represeted as poits o the (a,b) plae).

4 (,b) (a,b) (,) (a,) Biomial Theorem: The vectors are ot orthogoal, but the iteractio term is represeted by a outer product a b, so that the idepedet compoets of the vector is still represeted by the orthogoal bases ( ( i, j ) : c k ( ai b j) ( ai b j) a ( i j) ab( i j) b ( i j) ( a ab b ) k, ad c a ab b. Note that the outer product has resulted i a ew dimesio (umber lie, set) desigated by the uit vector k Note that both outer products are positive if oe uses the right had rule ccw from the horizotal axis ad the left had rule cw from the vertical axis, with the iteractio term ab

5 = rem(a,b,) k risig out of the page. (a,b) (,) (a,)

6 Arithmetic Operatio Elemets remai o the same umber lie (i the same set). For positive itegers (a,b,c): ( a b) i ai bi ci, c a b ai bi ( ab) i Vector Operatios (two dimesioal spaces ad up) No iteractio product: (=) )Pythagorea triples) betwee idepedet elemets of differet sets (umber lies) are represeted by ier products whe the resultat is calculated. Ier products betwee elemets i the same set result i a zero resultat: a b ai b j ( ab)( i j) for i ad j orthogoal Iteractio Product (=): (biary multiplicatio) betwee idepedet elemets of differet sets are represeted by outer products, ad the resultat is a positive umber lie (a psuedovector) i a ew dimesio, where the resultat is represeted by the sums of squares of scalars represetig the elemets i each set as before, but icludig the iteractio term as a biary product. The uit vector for the ew dimesio is positive ad orthogoal to the plae (i,j), formig a three dimesioal volume represeted by vectors costructed o the basis ( i, j, k ), where the scalar o the l.h.s. is depedet o the idepedet scalars o the r.h.s. for i ad j orthogoal ck ai b j ( ab)( i j) Therefore, for the case =, the form is EITHER a Pythagorea Triplet (o iteractio terms betwee biary compoets of the idepedet idepedet sets) OR the Biomial Theorem applies (iteractio terms give by biary multiplicatio of the compoets where the particle couts are preserved by set idepedece), but ot both This result ca the expaded to powers of c for > by the Biomial Theorem for higher powers (eergy/particle cout is ot coserved, sice there are o Pythagorea triples for > :,,, where c a b a b rem a b Therefore the relatio rem a, b, by the Biomial Theorem c a b caot be true; c a b, sice rem(a,b,) is ALWAYS positive ad > for positive particles/itegers. c a b for itegers (a,b,c) for > is Fermat's hypothesis QED

7 The oly relatio that preserves the total umber of particles (ad thus coserves eergy) is that of a Pythagorea triple (which satisfies the triagle equality). This result ca be expressed as a vector, with a temporary questio mark idicatig the ukow vector basis, which will deped o the elemets of rem(a,b,) cr a i b j rem( a, b, ) k, where r is the resultat vector. Agai, the positive orietatio of rem(a,b,) is maitaied by usig the right had rule for for the horizotal axis ( a ) ad the left had rule for the vertical axis ( b ) (,) The motivatio for this represetatio is that multiplicatio withi a sigle set (of positive itegers with a metric) i the space (a,b) ca be represeted by ier ("dot") products o a sigle umber lie (dimesio), where the agle betwee each pair of elemets is, so the products are scalars aliged alog a sigle uit vector. The vector alog the i axis with coefficiet to the power is represeted by ( a,) ; e.g. 5 ( a,) where 5 5 a i a a a a a i, ad similarly for the j axis with coefficiet b. Similarly, multiplicatio (iteractio) betwee two disjoit sets (a,b) are represeted by outer ("cross") where the agle betwee the elemets is / so for a sigle pair the operatio is represeted by: ck ai bj so that: m m m c k a i b j a b ( i j) ;

8 e.g. ( a a a) i ( b b b b) j ( a i b j a b ( i j) a b k The outer product creates a ew dimesio (idepedet umber lie; "pseudovector") orthogoal to the plae represeted by i, j. This ca be carried out successively, each time creatig a ew idepedet umber lie to a multidimesioal object (a hypervolume). Note that (e.g.) a b k a b k ( a b a b ) k. That is, for itegers r ad s: a r b s k a s b r k ( a r b s a s b r ) k These terms are the are the summed accordig to the Biomial theorem to yield the idividual terms [ rem rs, ( a, b)] k i rem( a, b) k. These terms form the elemets of the remaider term i the Biomial Theorem rem( a, b) k which is always positive ad ever vaishes, so Fermat's Theorem is proved for the geeral case. Rem(a,b) ca be thought of a "o-liearity" term. Cosider the case =. Here rem(a,b) = ab which is preset i the cases distict from the Pythagorea triples. I the case of Pythagorea triples, the resultat ca be thought of as a metric for each pair (a,b); the "distace" from the origi (,) to the pair (a,b) i the plae defied by i, j. I this case, a ad b are orthogoal, so ( ab ) =. If there is o such resultat, the Biomial Theorem is valid; that is, rem(a,b) exists, but is ot cotaied i the set represeted i the set {c} = {a} x {b} = (a,b) Sice {c} exhausts the relatios betwee the two sets of positive itegers {a} ad {b}, it is a complete descriptio of the set relatios ad operatios relevat to Fermat's Theorem whe exteded to higher powers of by the Biomial Theorem. Recap ad commet I the case of Pythagorea triples for the case =, the reaso that the iteractio term vaishes is because the particles do't iteract. Sice they are represeted by legs of a right triagle, the dot (ier) product vaishes. The other cases are characterized by the Biomial Theorem ad the cross (outer) product: Pythagorea Triple, ( = ) [ ai bj ] ab( i j) ab() Biomial Theorem, ( = ):

9 [ ai bj ] ab( i j) ab(1) ab For higher orders of ( > ), there is always a rem(a,b,) > term (from the Biomial Theorem) because of the rem( a, b, ) k vector, which derives from the cross products, so Fermat's Theorem is prove for >. ( c ) r 1 ( a ) i ( b ) j rem( a, b, ) k ( a ) i ( b ) j where ( c ) r 1 is the resultat vector i dimesio (+1) - ot orthogoal or parallel to the basis vectors ( i, j ) withi the plae (=). This relatio is true for all > ad positive iteger sets {a}ad{b}. Although ( c ) r 1 is ot idepedet of the (i,j,k) volume, it is a complete umber lie/set by itself, ad is idepedet ay other sigle umber lie sice the resultat coefficiet set of each iteratio is geerated by the previous set, ad so cotais all its elemets. Sice all itegers are represeted i the iitial relatio c = a + b, ad the sets are geerated by the relatio: ( 1) ( 1) ( c ( a b) ( a b)( a b) ), all the itegers are icluded for >, ad eve for = if oe icludes the Pythagorea Triples. Therefore, ca set ( c ) r ( c ) r ( b ) j rem( a, b, 1) k ( ( ) ( ) (,, 1) a i b j rem a b k 1 1 ( ) ( ) a i b j 1 by simply relabelig the variables, where we have simply relabeled the coefficiet of the resultat of the previous power of +1 to represet that represeted by the relabeled uit vector i Fermat's Theorem (the iequality) is verified for the ext higher dimesio, ad thus for all other higher dimesios by iductio. This observatio coceptually allows the orthogoality of costituet idepedet variables to be reset iteratively for each higher dimesio geerated by the outer products of the previous expasio. QED

10 I wat to stress oce agai (sigh) that Fermat's Theorem itself requires that >, the case = is irrelevat to the proof. ================================================== The questio is: Is the Biomial Theorem valid for positive itegers is valid for the case = eve for Pythagorea Triples? The aswer is "o"; i the case of Pythagorea Triples, "c " ad "a + b " refer to the same iteger (the "resultat", aalogous to the radius of a circle i real umbers). The reaso this is the case is that the resultat of the vectors i this case is still withi the (a,b) plae, ad ca be expressed as i the plae (a,b) c r a i b j where the resultat vector r remais That is, ca be expressed i terms of the sigle basis vector r area where c r a i b j ad the magitude is area area area relatio betwee areas). c ( a b ) (subscripts emphasize the This meas that the resultat is i effect a third "umber" lie withi the plae with the legth c ad slope or b. b a The magitude of this vector a b, so the form of the equatio remais the same for ay choice of a c a b, remais a iteger - however the symbol c ad the symbol refer to the same sigle positive iteger i the ew umber lie, represeted by c rarea a iarea b jarea where i area ad j area are orthogoal uit vectors that relate the uique areas that characterize the Pythagorea triple. This is the reaso Fermat's Theorem does ot apply to the case = for Pythagorea triples (i.e., the Biomial Theorem is does ot apply - the expressio is ot biomial). This is ot true of higher orders of, where there are iteractio terms, sice the resultat vector cr depeds o these iteractio terms that make up rem(a,b,) k. Therefore, is ot idepedet of iteractive products such as (a,b) for ay (a,b,) for ( c r a i b j rem( a, b, ) k. cr p q ab (i.e., ot embedded i the plae

11 Summary Cosider the followig sytactical expressio: c = a + b Note to reader: Before cotiuig, please read my igore list. What this expressio states i its simplest form is that the result is predetermied. For 6 = = 4 +, the expressio is merely a way of subdividig the elemets i a set i which there is a metric (e.g., 6 > 3). This ca be expressed i a umber lie related to the uit elemet where c 1 a 1 b 1 ( a b) 1 (The Distributive Law) Oe ca thik of the uity elemet as the basis for relatig all other elemets i the set. This meas that the variables "a", "b", ad "c" are sytactical elemets that show a operatioal relatioship; i the above case, oe of subdividig the set i such a way that the fuctio "+" relates the elemets a ad b to the fial elemet c, but the left ad right had sides of the equality refer to the same umber: c = c ad c = (a+b) The sigle umber lie ca be expressed as a matrix of oe dimesio: c = a+b = a + b Now cosider the case where we begi with a sigle umber lie {set} desigated by a ad add a secod umber lie represeted by b. {a} -> {a} +{b} Sice each set {a} ad {b} each cotai all possible elemets, a distictio must be made betwee the sets. This is doe by Cartesia coordiates (Cartesia products), where the third elemet {c} ow refers to the resultat set created by the relatioship betwee the two distict sets {a} ad {b}. The disjoit sets (e.g., positive itegers) {a} ad {b} are ofte expressed as a ordered pair (a,b). For real umbers, a ad b are replaced by x ad y i the ordered pair (x,y). That is, the elemet {c} does ot belog i either {a} or {b} if a ad b are disjoit (idepedet) sets. For set theory, this is expressed by the Cartesia Product: C A B For vectors, this is expressed by the vector otatio: cr ai bj where i ad j are uit vectors i each distict set, ad r is the resultat that belogs i either set; a, b, ad c are

12 called scalars. I matrix otatio, the distict sets {a} ad {b} have the represetatio: a b, ad the resultat of the operatio "+" (ow vector additio) is represeted by c. The arithmetic operatio for the "legth" (vector magitude) of the resultat satisfies the relatio c a b, (the triagle equality), which is complete sice there are o other sets or elemets thereof. If a ad b are positive itegers, this meas that the Biomial Theorem is a vector relatio i two dimesios, ot a arithmetic relatioship i a sigle dimetio, sice rem(a,b,) are elemets that ot i either set { a } i ad { b } j so Fermat's Theorem is prove. That is, Note that rem(a,b,) caot reside i either {a} or {b} sice it cosists of iteractive elemets of both sets. It is impossible to prove Fermat's theorem i oe dimesio, sice the sum (a + b) ad c ca refer to the same uique positive iteger, ad likewise for the expressio d c a b

13 Relatio to SU() Oe might thik that this result ca be exteded to SU(): a rem( ab) a rem( ab) a rem( ab) b b rem( ab) b (because rem(a,b) > ) (where the Pauli matrix: yields: So particle umber is ot coserved for > (ad ot eve for = if the form is ot a Pythagorea triple). 1 rem( a, b) rem( a, b) 1 rem( a, b) rem( a, b) The resultat of the vector o the r.h.s. is equal to rem(a,b). Note that trace 1 det, but that ( 1) (1 ) 1, where ( ), ad 1 1 Tr( ) is the x idetity matrix where ad det( ) 1. 1 The matrix 3 1 represets a pair massless electros with positive or egative spi (ot charge, sice they are both electros) before the B field is tured o i the Ster-Gerlach experimet (the iitial state).

14 1 i The matrices 1 ad each represet pairs of these electros at the fial 1 i state; represets the pair of electros after the B field has bee tured o, ad 1 represet the B field itself. These matrices result from the part of the Lortez force equatio E k vi B j ( vb) k (Note that the traces of the matrices are all ) vb

15 For the case = (Biomial Theorem): A a, B b C A B a b Tr( C) a b c a a a a B ( A ) b b b b Tr( A B) a b c For the Biomial Theorem (case = ): a ( A B) b A B AB it ab Tr ( A B) it = a b ab For the Biomial Theorem (case > ) C a b rem( a, b, ) Tr( C )= a b rem ( a, b, )

16 For the Biomial Theorem (case = ) we have the idepedet matrices A ad B ad the iteractio matrix: a a b a ab b a b b ab TAB ( A B ) AB Tr T a b ab ( AB) For completeess (case > ) oe might thik that the iteractios ca be characterized i SU() by: (,, ) a rem a b 1/ T AB ( A B ) R em( AB) b rem( a, b, ) 1/ a rem( a, b, ) b rem( a, b, ) where half of the iteractio term has bee assiged to the off-diagoals. The rem( a, b, ) Tr( T AB) a b ( ) a b rem( a, b, ) The problem with this approach is that it treats the iteractive terms as appearig o the a ad b umber lies as o-itegers, istead of icreasig the dimesio of the vector space (with half of sqr(rem(a,b,)) assiged to each umber lie i first order ad thus oitegers o those umber lies. The difficulty with o-iteger elemets i the trasitios is resolved by simply assigig the sums ad products or rem(a,b,) term by term to a third dimesio as i post #488, where k i j j i, where the outer products commute so that k 1. Here the outer products correspod to ordiary multiplicatio of positive itegers (which commute) so that all terms remai positive, as i the Biomial Theorem for positive itegers.

17 Relatio to Special Theory of Relativity Case ( vc / ) 1, ( t'/ t) 1 (This sectio will be updated shortly to iclude a discussio of the Loretz Trasforms. To see the directio, see my web articles) Relativity The fial state ct is greater tha the iitial state ct via a perturbatio vt. The Time Dilatio Equatio is show i the followig diagram, where ct is the iitial state, ct is the fial state,, ad vt is a perturbatio (chage of state). The first particle is geerated by the relatio 1 dv dv = c of the horizotal axis. Multiplyig by m ct gives the rest mass of the first particle. c c v, ad is the basis vector (Note to GTR ethusiasts the chage i v from v= to v=c is actually a acceleratio ; similarly, the chage from ct -> ct is also a acceleratio, ad for t,t is a scalig factors that ca also be labeled c t, so a chage i the speed of light from the iitial coditio ct)

18 Dividig by ct gives yields the relatio i terms of ad, which form orthogoal axes for v/ c 1 ad t'/ t 1 : Subsequet particles are geerated by the perturbatio vt ' which is a ccw rotatio of the horizotal axis to the vertical axis util vt ' ct ' at which poit the ew ad the secod particle is represeted by a secod dimesio (the vertical dimesio). This ew dimesio ca the be cotracted by rotatig it cw ad addig it to the horizotal axis to represet a ew ( ct '). Thus icreasig the particle cout (if the procedure is parameterized by t ad exteded aroud the whole circle, the fuctio ca the be 1 vt ' 1 1 si cos where si ( ) si ( ), so that ct represeted by (1 ) (si cos ) represets the creatio of particles. That is, each particle is created by a complete rotatio. For STR, the total eergy is represeted by ( m' c 1 ) (si cos ), where v ( m' c ) Pc ( mc ), P m'( ) c ( m ( 1) ) c ( m c )( ( 1) ) at ct ' ct 1 ad c ( 1) 1 (Or somethig like that I m workig at gettig clearer). Note that there are o itegers represeted by or except i the case v = c where v/c = 1, ( or ) = 1. (Note: i quotes, because the deomiator goes to zero whe v/c = 1, which meas the dimesio represeted by that deomiator o loger exits. (cosider divisio whe z = x/y, with as the midpoit of all possible legths i that dimesio, so is a poit, ot a legth. A dimesio i the real (or iteger) umbers requires a metric (legth).

19 Therefore, at y = (as a legth), the dimesio represeted by y does ot exist, ad z = x, which, i the case of = 1

20 Case ( vc / ) 1, ( t'/ t) 1 The fial state is greater tha the iitial state where the iitial state is the uit circle 1 (thik ct as the iitial state extedig to ct as a fial state, with 1 = ct I this case (the cotravariat Loretz trasform) the fuctio becomes hyperbolic (eergy is beig absorbed). The iteger plae (a,b) ca be superimposed o the positive quadrat to represet positive itegers as relativistic particles. (with obvious implicatios for Quatum Field Theory w.r.t. the other quadrats) (Update: i the diagram below, the equatio should actually be the Loretz trasform i the (cotravariat) frame x ' ( x vt) I ll try to fix the diagram shortly (the scalig for v/c is c c off a bit as well, ad the curve should be offset from the circle. This is a work i progress I also eed to figure out how to crop the dam thig.). I ll leave the diagram i util l ca correct it, though, as it does show the directio of the aalysis. This is the result of extedig the iteractio terms (cross-diagoal elemets) i the Loretz trasformatio matrix for v/c >1 at v = c the system defies a ew ct', ad a ew particle is created. At the limit, vt' -> ct, so the iitial coditio ct = h becomes ifiitesimal compared to the total eergy of the complete system' For STR, the process is: t t t t t t 1 t t 1 t t 1 t t 1 where vc / ad 1/ 1 Here is the mass creatio rate relative to c. The matrix:

21 represets the iteractio matrix, where v starts at v =, ad rus to v=c, at which poit a ew ivariat particle is created. At that poit the deomiator of becomes, which meas the dimesio represeted by the deomiator i terms of v/c o loger exists, ad the matrix o the left (the iitial state) is icreased by oe (a ew particle has bee created So the terms perform a aalogous fuctio to rem(a,b,n) o a term by term basis, ad is why Fermat's Theorem is relevat to particle physics (by cotrast).

22 ( 1) 1 gives the samplig (iteger, particle) cout as 1 (i.e.,), icreases (or somethig like that.) Note: The graph is somewhat misleadig, sice is shows egative (imagiary ( i ) ) values (fermios) i additio to positive defiite bosos. This is the additio to the aalysis of Dirac, ad is a precursor of thigs to come.

23

24 Trasitio for >= Trasitio: vt ' ct ' ct 1

25 Note that the iitial coditio ( 1 ) is replaced by the ivariat h (Plack s costat) becomes progressively isigificat as icreases. If the iitial coditio is replaced by h Plack s costat, (i.e., as as, ) vt ' That is, as vt ' ct ', 1, the effect of h relative to the total mass approaches, ct ' which is the limit of (o-quatum mechaics) classical physics.

26 The equatio is liear i, so that gauge ivariace is preserved: The iitial creatio of the uiverse (as a particle) ca be represeted by the process (loosely): vt ct m 1 U v / c ct m 1 The secod particle is created by (loosely): U() ct ct vt ' ct m 1 vt ' ct m 1 ct ct vt ct 1 vt ct 1 (I m ot altogether sure how to express the followig pg.; I ll keep workig o it.. For positive particle (boso) cout to be coserved, the factors (,, ) must be a v Pythagorea triple for 1 c (that is, c 1 1 so the deomiator of is ot imagiary. This is impossible for idetical (idistiguishable) particles (a,a) sice the mass will always be a factor of at v/c =1. (i.e., " " 1, 1 each time a ew particle is created (a,a, a

27 The particle is actually created alog the lie of 45 degrees ( /4), so that each cw (from /) ad ccw (from ) rotatio gets ½ the mass. Of course, symmetry ca be applied i all four quadrats for other tha positive iteger particles. If the eergy is absorbed, the the ew particle ct is icreased, but if a distict particle is created i.e., the source is idepedet of the origial mass), the a ew dimesio is created, reflectig the ew particle cout. Subsequet perturbatios will still be reflected i the origial two dimesios, with the particle cout (full rotatio, iteger boost) reflected by the dimesio of the matrix. The process of icremetig particles is: ct, vtdvdt ct ct ct ' Icremetig the particle cout i the positive quadrat by rotatio produces the boost: Repeatig the process times yields: Where each successive icremet is characterized by: The trace of the matrix is the magitude of the resultat: The triagle iequality yields: N 1, ad the trace yields the magitude of the vector: Tr( N ) 1 (as goes to ifiity, the iitial coditio becomes isigificat).

28 M m, ad the trace yields the magitude of the vector: m Tr( M ) m m (as goes to ifiity, the iitial coditio becomes isigificat). The eergy of the first uiverse (as a positive iteger) ca be destroyed by usig a complex umber: ( 1 i ) (1 1) (Bottom lie tbd ) The Biomial Theorem together with Fermat s Theorem tells oe how to cout additioal sets of Uiverses. For two Idetical Uiverses, if they do ot iteract, the STR is valid. If they do ot iteract (ad preserve their itegrity - i.e., they both have the same value for c ; the STR holds. If they are differet, ad idetifiable as such, the they must have differet values for c (so differet debroglie wavelegths / Plack s costat, etc.), but distictio is maitaied by the Biomial Theorem.

29 Addedum (from the forum): Whe oe makes the statemet "for all positive itegers a ad b", the to make a distictio betwee them, they must belog to disjoit (distict) sets. That is, whe the sets are disjoit, they do ot "iteract"; there is o relatio betwee the elemets. This is the case for Pythagorea triples, where the legs of the triagle are perpedicular, so the dot product vaishes. (for grapes): For the case = (Biomial Theorem) ab does ot vaish because the legs of the triagle are ot perpedicular. However, ab is a area, ad is i the same plae as the ordered pair (a,b). So Rem(a,b,)=ab. The case of is ot relevat to the proof of Fermat's Theorem ( > ) For >, the terms i rem(a,b,) have the geeral form of the Biomial theorem (e.g. 3 3 a b a b ). I used "absolute value" of cross products merely to idicate positive iteger multiplicatio, but also that these products ca ever be i either {a} or {b}, so therefore could ever be i the plae (a,b). Therefore they ca be represeted as a "o-iteractig" vector w.r.t (a,b) by a third vector: { a } i { b } j} { a } i { b } j} { a b } { a b } k { a b a b } k The sets i the Biomial Theorem are the related by vector additio, with the resultat vector ot orthogoal to either i, j, or k, but a mixture (e.g., the hypoteuse of the right triagle for = (Pythagorea) - or the logest leg of the triagle for Biomial case =. The case of is ot relevat to the proof of Fermat's Theorem ( > ) For >, the the full expressio for the Biomial Theorem is represeted by: { c } r { a } i { b } j { rem( a, b, ))} k, which holds for all members of the sets. Sice o elemet i {rem(a,b,)} =, the the sets a or b so Fermat's Theorem is proved. cr is ot i the plae(, ) a b, much less i

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