tag:blogger.com,1999:blog-12535639.post115025213850624479..comments2024-02-26T06:55:41.876-08:00Comments on Fermat's Last Theorem: Another False Proof: E. E. EsculturaLarry Freemanhttp://www.blogger.com/profile/06906614246430481533noreply@blogger.comBlogger32125tag:blogger.com,1999:blog-12535639.post-60345032340764113202013-07-07T05:36:43.008-07:002013-07-07T05:36:43.008-07:00Correction.
FLT cannot be resolved in the real nu...Correction.<br /><br />FLT cannot be resolved in the real number system but has been re-formulated and resolved in the new real number system.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-1716539299244049202013-07-06T23:12:12.696-07:002013-07-06T23:12:12.696-07:00Reaply to Maciej Ma
FLT is a deeper problem that...Reaply to Maciej Ma <br /><br />FLT is a deeper problem that what it seems. It is posed as a problem in the system of integers or the real number system that includes the former as a subspace neither of which is well defined. In fact, the real number system is inconsistent, counter examples to its trichotomy and completeness axioms having been constructed. Therefore FLT is ambiguous and cannot be resolved. <br />For complete discussion of these issues and the resolution of FLT see, Escultura, E. E., The new real number system and discrete computation and calculus. J. Neural, Parallel and Scientific Computations, 17, 2009, pp. 59 - 84Moreover, complete discussion of these issues are posted on this wibsite. E. E. EsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-25822728537914892942013-07-03T05:16:41.090-07:002013-07-03T05:16:41.090-07:00FERMAT's LAST THEOREM
X ^n + Y^n = Z^n , X,...FERMAT's LAST THEOREM <br /><br />X ^n + Y^n = Z^n , X,Y,Z and n are Natural Positive ( N+) , and n >2 <br /><br />MAROSZ' s conditions ( couterexample ) <br /><br />X= [2^(n/n)] , Y = [2^n/n] , Z = [2^n+1/n] and n=3 <br /><br />n=3 > 2 [ok ]<br /><br />2^(n/n) = 2 it is ( N+) , [ok ]<br /><br />2^(n+1)/n = 2^(4/3) = 16^(1/3) = 4^(1/2) = 2 it is N+ [ok]<br /><br />Fermat's equation and Marosz's conditions <br /><br />basic equation : [2^(n/n)]^n + [2^n/n]^n = [2^n+1/n]^n <br /><br />LEFT <br />[2^(n/n)]^n = 2^ [(n/n) *n] = 2^n <br /><br />2^n + 2^n = 2*(2^n) , n=3 2* (2^3) = 16 <br /><br />RIGHT <br />[2^((n+1)/n)]^n = [2^(4/3)]^3 = 2^4 = 16 <br /><br /><br />authore <br />http://maroszmaciej.blogspot.com/Anonymoushttps://www.blogger.com/profile/10501701212436630364noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-45049841618202869262010-11-23T05:40:40.768-08:002010-11-23T05:40:40.768-08:00Reply to Kevin,
You are rather late, Keven, but b...Reply to Kevin,<br /><br />You are rather late, Keven, but before you pull out something from the flat of your foot read the original version of my counterexamples to FLT in:<br /><br />Escultura, E. E., The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations (Dynamic Publishers), 2009, Vol. 17, pp. 59 – 84.<br /><br />Read and refute my arguments point by point.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-60278563805705904922010-11-09T16:50:45.327-08:002010-11-09T16:50:45.327-08:00Wow... What an awesome example of a crank. I have ...Wow... What an awesome example of a crank. I have never seen someone go on such a diatribe about absolute nonsense. I have posted concrete arguements against Esculturas number system and 'disproof' of FLT. He fails to respond to any onf them, but then says nobody has ever argued convicingly against him. I am far from the only one, there are many blogs with irrefutable proof that Escultura is incorrect. However, instead of arguing against their points, he just posts a seemingly never ending series of comments where he does nothing but repeat himself over and over, never once giving any proof that their arguments against him are incorrect.<br /><br />What in insecure little man!kSaugahttps://www.blogger.com/profile/10383064863774679836noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-2174872555274447902010-04-27T02:35:21.958-07:002010-04-27T02:35:21.958-07:00CALL FOR A GRAND UNIFIED JOINT CELEBRATION
Materi...CALL FOR A GRAND UNIFIED JOINT CELEBRATION<br /><br />Materialist philosophers of all cultures must have pondered this question: what are the basic constituents of matter? The Greeks answered it with four elements they found in nature: earth, water, fire and air. The Chinese added one more item – wood. Of course, they were not satisfactory and since then the search for the basic constituent of matter was in limbo for 5,000 years until in the 1950s inspired by the exciting development of quantum physics particle physicists renewed the search with vigor by smashing the nucleus of the atom in pursuit of the basic irreducible elementary particles or building blocks of matter. By the 1990s the search was a complete success with the discovery of the +quark (up quark) and quark (down quark) and the electron (discovered in 1897). They are basic as constituents of every atom; a heavy isotope has one more constituent – the neutrino. The particle physicists have, indeed, found what they were looking for – the irreducible building blocks of matter – and whatever they have found beyond these is a bonus for natural science. <br /><br />In the 1980s dark matter came to the fore with overwhelming evidence of its existence [6,7,8] and, using the new methodology of qualitative modeling that explains nature and its appearances in terms of natural laws [1,5], was established in 1997 [4] as one of the two fundamental states of matter the other being ordinary or visible matter [2,5]. That same year the building block of dark matter, the superstring, was discovered as the crucial factor for the solution of the gravitational n-body problem [4] and development of the grand unified theory (GUT). The latter has been established in a series of papers since 1997 and consolidated in [2]. There is only one basic constituent in view of the non-redundancy and non-extravagance natural principles [3] just as there is only one electron since all electrons have identical structure, properties, behavior and functions and differ only in locations. Moreover, it was also established that the superstring coverts to the basic elementary particles as agitated superstring [1,2,3]. In effect, this established the superstring as the basic constituent of matter, dark and visible <br />[1,2,3,4,5]. <br /><br />This happy turn of events came without fanfare and even without notice. It is an important milestone for science and calls for a grand unified joint celebration by particle and theoretical physicists to mark these monumental achievements and the threshold of a new epoch for natural science and its applications. It is even worth a world congress of particle and theoretical physicists.<br /><br />References<br /><br />[1] Escultura, E. E., The mathematics of the grand unified theory, Nonlinear Analysis,<br />A-Series: Theory: Method and Applications, 71 (2009) <br />e420 – e431.<br />[2] Escultura, E. E., The grand unified theory, Nonlinear Analysis, A-Series: Theory: Method and Applications, 69(3), 2008, 823 – 831.<br />[3] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, Nonlinear Analysis, B-Series: Real World Applications, 11 (2009),<br />29 – 38.<br />[4] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear Analysis, A-Series: Theory, Methods and Applications, 38(8), 521 – 532.<br />[5] Escultura, E. E., Superstring loop dynamics and applications to astronomy and biology, Nonlinear Analysis, A-Series: Theory: Method and Applications, 35(8), 1999, 259 – 285.<br />[6] Astronomy (a) August 1995, (b) January 2001, (c) June 2002.<br />[7] Science, Glow reveals early star nurseries, July 1998.<br />[8] Science, (a) Starbirth, gamma blast hint at active early universe, 282(5395), December, 1998, 1806; (b) Gamma burst promises celestial reprise, 283(5402),<br />January 1999; (c) Powerful cosmic rays tied to far off galaxies, 282(5391), Nov. 1998, 1969 – 1971.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-64830707623029058652010-03-09T15:36:09.902-08:002010-03-09T15:36:09.902-08:00Thank you very much for your reply .
Ahmed Idriss...Thank you very much for your reply .<br /><br />Ahmed Idrissi BouyahyaouiAhmed Idrissi Bouyahyaouihttps://www.blogger.com/profile/05028445932088862369noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-59964537862238302572010-02-28T19:25:38.380-08:002010-02-28T19:25:38.380-08:00Correction to my previous post:
I constructed cou...Correction to my previous post:<br /><br />I constructed countably infinite counterexamples to FLT. EEEsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-89721385703078780102010-02-28T19:21:47.800-08:002010-02-28T19:21:47.800-08:00This is my reply to
Ahmed Idrissi Bouyahyaoui'...This is my reply to <br />Ahmed Idrissi Bouyahyaoui's comments.<br /><br />1) I did not prove FLT; on the contrary, I disproved it by counterexamples.<br /><br />2) Using standard mathematics I do not find a flaw in your argument.<br /><br />3) My main point is that the field axioms of the real number system are inconsistent; therefore, this number system is ill-defined; consequently, FLT being formulated in it is also ill-defined, ambiguous and ill-formulated as a problem.<br /><br />3) Therefore, I constructed the consistent new real number system R* using the elements 0, 1 well defined by the addition and multiplication tables (as the two other axioms).<br /><br />4) Then I constructed the countable counterexamples to FLT in R*.<br /><br />Thank you for the comment.<br /><br />E. E. EsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-63283056914610964702010-02-28T14:46:04.817-08:002010-02-28T14:46:04.817-08:00I'm sorry, the "proof" given above i...I'm sorry, the "proof" given above is insufficient. <br />I propose what I believe to be sufficient proof. This proof has two parts: <br />1) classical logic, <br />2) arithmetic <br /><br />www.happy-arabia.org/FLTproof.pdf <br /><br />Ahmed Idrissi BouyahyaouiAhmed Idrissi Bouyahyaouihttps://www.blogger.com/profile/05028445932088862369noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-44114263820875259292010-02-15T19:14:01.750-08:002010-02-15T19:14:01.750-08:00I need your opinion. Thank you for your reply.
TH...I need your opinion. Thank you for your reply.<br /><br />THE FERMAT’S LAST THEOREM :<br />« z, y, x, n ϵ N*, n>2 : z^n ≠ y^n + x^n » <br />DIRECT PROOF : <br />Resolution method : recurrence by induction. <br /><br />The proposition :<br />(1) ( y, x, n ϵ N*, gcd(y,x)=1, n>2 : (1^n ≠ ^y^n + x^n))˄<br /> ( y, x, n ϵ N*, gcd(y,x)=1, n>2 : (1^n ≠ y^n - x^n))˄ <br /> ( y, x, n ϵ N*, gcd(2,y,x)=1, n>2 : (2^n ≠ y^n + x^n))˄<br /> ( y, x, n ϵ N*, gcd(2,y,x)=1, n>2 : (2^n ≠ y^n - x^n))˄<br /> ( y, x, n ϵ N*, gcd(3,y,x)=1, n>2 : (3^n ≠ y^n + x^n))˄<br /> ( y, x, n ϵ N*, gcd(3,y,x)=1, n>2 : (3^n ≠ y^n - x^n))˄<br /> ( y, x, n ϵ N*, gcd(4,y,x)=1, n>2 : ((3+1)^n ≠ y^n + x^n))˄<br /> ( y, x, n ϵ N*, gcd(4,y,x)=1, n>2 : ((3+1)^n ≠ y^n - x^n)) ,<br />gives the first three and the fourth elements of a recurrence.<br /><br />By hypothesis : z^n ≠ y^n + x^n for z, y, x, n ϵ N* and n>2, the three numbers z^n ≠ y^n + x^n, y^n ≠ z^n - x^n and x^n ≠ z^n - y^n with z^n > y^n > x^n ≥ 1 .<br />So, it needs at least three consecutive elements to determine if the series with first elements (1) is inductive or not.<br /><br />Proof of (1) :<br />As any integer n>2 is multiple of 4 or odd prime, it suffices to prove<br />the Fermat’s last theorem for n=4 and for each odd prime. <br /><br />For n=4 :<br />2^4 = y^4 - x^4 = (y^2-x^2)( y^2+x^2) , impossible equality, the factor (y^2+x^2) , greater than 4, isn’t power of 2 (y^2+x^2 ≡ 2 mod 4).<br />3^4 = y^4 - x^4 = (y^2-x^2)( y^2+x^2) , impossible equality, the factor (y^2+x^2) , greater than 3, isn’t power of 3 (y^2+x^2 ≡ 2 mod 3).<br /><br />For n=p , p odd prime :<br />2^p = y^p – x^p <br />2=y-x mod p → y-x≠1 , 2^p = (y-x)[(y^p – x^p)/(y-x)] ,<br />the two factors of the second member are necessarily coprime, therefore the equality is impossible.<br />3^p = y^p – x^p<br />3=y-x mod p → y-x≠1 , 3^p = (y-x)[(y^p – x^p)/(y-x)] ,<br />for p≠3, the two factors of the second member are necessarily coprime, therefore the equality is impossible.<br />For p=3 :<br />3^3 = (y-x)[(y^3 – x^3)/(y-x)]= (y-x)(y^2+y*x+x^2), impossible equality, since<br />y^2+yx+x^2 > (y-x)^2 > y-x ≥ 3 → y-x = 3 , (y-x)^2= 32, y^2+y*x+x^2 > 3^2 .<br /><br />Given the proposition (1), suppose that for an integer r > 4 and for all integer z, 1 ≤ z ≤r, we have : <br />(2) ( y, x, n ϵ N*, n>2 : (z^n ≠ y^n + x^n))˄<br /> ( y, x, n ϵ N*, n>2 : (z^n ≠ y^n - x^n)).<br />In this hypothesis, we also have : <br /> y, x, n ϵ N*, n>2 : (r+1)^n ≠ y^n + x^n, <br />otherwise y, x, n ϵ N*, n>2 : (r+1)^n = y^n + x^n → y^n = (r+1)^n - x^n contrary to the hypothesis (2), <br />since (r+1) > y > x → x < y ≤ r → y^n ≠ (r+1n - x^n → (r+1)^n ≠ y^n + x^n.<br /><br />The principle of recurrence by induction allows to conclude : <br /> <br />The proposition y, x, n ϵ N*, n>2 : z^n ≠ y^n + x^n is true for z=1, 2, 3, 4, …... , r (r>4) and r+1, it is true for all z ϵ N* :<br />(3) z, y, x, n ϵ N*, n>2 : z^n ≠ y^n + x^n . <br /><br /><br /><br />Detailed explanation :<br /><br />Let P(z) = (z^n ≠ y^n + x^n)˄(z^n ≠ y^n - x^n) = P+(z)˄P-(z) .<br />As P(1) (then P+(1)), P(2) (then P+(2)), P(3) (then P+(3)), P(4) (then P+(4)) are true and, for r>4, P(r) (then P+(r)) implies P+(r+1),<br />the principle of recurrence by induction allows to conclude : <br />P+(r) is true for all r : y, x, n ϵ N*, n>2 : r^n ≠ y^n + x^n .<br /><br />Ahmed Idrissi Bouyahyaoui <br />© inpi – ParisAhmed Idrissi Bouyahyaouihttps://www.blogger.com/profile/05028445932088862369noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-37894910681112250112009-11-02T02:58:42.672-08:002009-11-02T02:58:42.672-08:00Like a nonterminating decimal, an element of d* is...Like a nonterminating decimal, an element of d* is unaltered if finite g-terms are altered or deleted from its g-sequence. When delta = 1 and a_1a_2…a_k = 1 (7) is called the basic or principal d-sequence of d*, its g-limit the basic element of d*; basic because all its d-sequences can be derived from it. The principal d-sequence of d* is, <br /><br />(0.1)^n , n = 1, 2, … (8)<br /><br />obtained by the iterated difference,<br /><br />N – (N – 1).99… = 1 – 0.99... = 0 with excess remainder of 0.1;<br /> 0.1 – 0.09 = 0 with excess remainder of 0.01;<br /> 0.01 – 0.009 = 0 with excess remainder of 0.001;<br /> ………………………………………………… (9)<br /><br />Taking the nonstandard g-limits of the left side of (9) and recalling that the g-limit of a decimal is itself and denoting by d_n the d-limit of the principal d-sequence on the right side we have, <br /><br />N – (N – 1).99… = 1 – 0.99... = d_n. (10)<br /><br />Since all the elements of d* share its properties then whenever we have a statement “an element d of d* has property P” we may write “d* has property P”, meaning, this statement is true of every element of d*. This applies to any equation involving an element of d*. Therefore, we have,<br /><br />d* = N – (N – 1).99… = 1 – 0.99... (11) <br /><br />Like a decimal, we define the d-norm of d* as d* > 0.<br /><br />We state some theorems about R*.<br /> Theorem. The d-limits of the indefinitely receding (to the right) nth d-terms of d* is a continuum that coincides with the g-limits of the tail digits of the nonterminating decimals traced by those nth d-terms as the aks vary along the basic digits. <br /> Theorem. In the lexicographic ordering R* consists of adjacent predecessor-successor pairs (each joined by d*); therefore, the g-closure R* of R is a continuum [9].<br /> Corollary. R* is non-Archimedean and non-Hausdorff in both the standard and the g-norm and the subspace of decimals are countably infinite, hence, discrete but Archimedean and Hausdorff. <br /> Theorem. The rationals and irrationals are separated, i.e., they are not dense in their union (this is the first indication of discreteness of the decimals) [7]. <br /> Theorem. The largest and smallest elements of the open interval (0,1) are 0.99… and 1 – 0.99…, respectively [6]. <br /> Theorem. An even number greater than 2 is the sum of two prime numbers.<br /><br />Remark. Gauss’ diagonal method proves neither the existence of nondenumerable set nor a continuum; it proves only the existence of countably infinite set, i.e., the off-diagonal elements consisting of countable union of countably infinite sets. The off-diagonal elements are not even well-defined because we know nothing about their digits (a decimal is determined by its digits). We state the following corollaries from our discussion: (1) Nondenumerable set does not exist; (2) Only discrete set has cardinality; a continuum has none.<br /><br />(This article is excerpted from Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 – 84)<br /><br />E. E. Escultura<br />Research Professor <br />GVP - V. Lakshmikantham Institute for Advanced Studies<br />and Departments of Mathematics and Physics<br />GVP College of Engineering, JNT University <br />Madurawada, Visakhapatnam. AP, IndiaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-45457900641845281242009-11-02T02:57:50.470-08:002009-11-02T02:57:50.470-08:00Now, we allow delta to vary steadily in its domain...Now, we allow delta to vary steadily in its domain and also the a_js along the basic integers (not simultaneously 0). Then their terms trace the tail digits of all the decimals and as n increases indefinitely they become smaller and smaller and indistinguishable from each other. We call their nonstandard limits dark numbers and denote by d* which is set valued, countably infinite and includes every g-limit of the nonstandard d-sequence (7). To the extent that they are indistinguishable d* is a continuum (in the algebraic sense since no notion of open set is involved). Thus, the tail digits of the nonterminating decimals merge and form the continuum d*.<br /> At the same time, since the tail digits of all the nonterminating decimals form a countable combination of the basic digits 0, 1, …, 9 they are countably infinite, i.e., in one-one correspondence with the integers. In fact, any set that can be labeled by integers or there is some scheme for labeling them by integers is in one-one correspondence with the integers, i.e., countably infinite. It follows that the countable union of countable set is countable. Therefore, the decimals and their tail digits are countably infinite. However, as the nth d-terms of (7) trace the tail digits of the nonterminating decimals they become unidentifiable and cannot be labeled by the integers anymore; therefore, they are no longer countable. In fact they merge as the continuum d*.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-22929153168533164322009-11-02T02:55:43.182-08:002009-11-02T02:55:43.182-08:00As we raise n, the tail digits of the nth g-term o...As we raise n, the tail digits of the nth g-term of any decimal recedes to the right indefinitely, i.e., it becomes steadily smaller until it is unidentifiable. While it tends to 0 in the standard norm it never reaches 0 and is not a decimal since its digits are not fixed; ultimately, they are indistinguishable from the similarly receding tail digits of the other nonterminating decimals. In iterated computation when we are trying to get closer and closer approximation of a decimal the tail digits may vary but recede to the right indefinitely and become steadily smaller leaving fixed digits behind that define a decimal. We approximate the result by taking its initial segment, the nth g-term, to desired margin of error. <br /><br />Consider the sequence of decimals, <br /><br />(delta^n(a_1a_2…a_k), n = 1, 2, …, (7) <br /><br />where delta is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a_1, …, a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (7) d-sequence and its nth term nth d-term. For fixed combination of delta and the a_j’s, j = 1, …, k, in (7) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (7) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < d < x.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-79920615949667614122009-11-02T02:53:32.508-08:002009-11-02T02:53:32.508-08:00Since addition and multiplication and their invers...Since addition and multiplication and their inverse operations subtraction and division are defined only on terminating decimals computing nonterminating decimals is done by approximation each by its nth g-terms (called n-truncation) and using their approximation to find the nth g-term of the result as its approximation at the same margin of error. (Note that the g-nth term is a terminating decimal whose last digit is the nth digit) This is standard computation, i.e., approximation by decimal segment at the nth digit. Thus, we have retained standard computation but avoided the contradictions and paradoxes of the real numbers. We have also avoided vacuous statement, e.g., vacuous approximation, because nonterminating decimals are g-limits of g-sequences which belong to R*. Moreover, we have contained the inherent ambiguity of nonterminating decimals by approximating them by their nth g-terms which are not ambiguous being terminating decimals. In fact, the ambiguity of R* has been contained altogether.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-87887801051294183582009-11-02T02:52:37.294-08:002009-11-02T02:52:37.294-08:00A sequence of terminating decimals of the form,
...A sequence of terminating decimals of the form, <br /><br />N.a_1, N.a_1a_2, …, N.a_1a_2…a_n, … (5)<br /><br />where N is integer and the a_ns are basic integers, is called standard generating or g-sequence. Its nth g-term, N.a_1a_2…a_n, defines and approximates its g-limit, the nonterminating decimal, <br /><br />N.a_a_2…a_n,…, (6)<br /><br />at margin of error 10^-n. The g-limit of (5) is nonterminating decimal (6) provided the nth digits are not all 0 beyond a certain value of n; otherwise, it is terminating. As in standard analysis where a sequence converges, i.e., tends to a specific number, in the standard norm, a standard g-sequence, converges to its g-limit in the g-norm where the g-norm of a decimal is itself. Note that a nonterminating decimal is well defined by its g-sequence although it is ambiguous.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-75833616054056425722009-11-02T02:50:28.472-08:002009-11-02T02:50:28.472-08:00THE FINAL STRETCH IN THE CONSTRUCTION OF THE NEW R...THE FINAL STRETCH IN THE CONSTRUCTION OF THE NEW REAL NUMBER SYSTEM R*: WELL DEFINING THE NONTERMINATING DECIMALS (for the first time)<br /><br />First we note that since a decimal is defined by its digits the only well defined decimals are the terminating ones. Nonterminating decimals are ill-defined or ambiguous because not all their digits are known. Therefore, the concept rational (and also irrational) is ambiguous because it is impossible to verify if its decimal representation is periodic since we cannot check all its digits being infinite. However, ambiguity can be contained by approximating it with certainty, e.g., by a terminating decimal (which has no ambiguity); such approximation is valid if the margin of error is known and can be made small as desired. Thus, while nonterminating decimals cannot be well defined we can contain its ambiguity to the point where we do algebraic operations with them and approximate the result with desired margin of error. Now we introduce the generating or g-sequence and its g-limit, a nonterminating decimal which has contained ambiguity (approximable by certainty).E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-78838261365065976042009-10-13T20:15:09.361-07:002009-10-13T20:15:09.361-07:00What is the difference between music and mathemati...What is the difference between music and mathematics?<br /><br />Answer: Classical music is a treasure but classical mathematics is a trash.E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-23625850386859558972009-09-17T05:40:29.146-07:002009-09-17T05:40:29.146-07:00Being a newcomer in mathematics and science (my fi...Being a newcomer in mathematics and science (my first publication is “The solution of the gravitational n-<br />body problem”, Nonlinear Analysis, Series A, 30(8), Dec. 1997, 521 – 532), I was an underdog in contending ideas with the heavy weights of science and mathematics. Therefore, I had to disseminate my ideas broadly as quickly as possible. The first forum I posted my message in was SciMath, 1997, and the topic was the equality 1 = 0.99… I said that this was really nonsense and I’ll explain why later. <br /><br />There was a howl of protest and hundreds of messages were posted in protest during the year. Some called me crackpot, lunatic, moron, etc. One even wrote my colleague (who sort of discovered me in mathematics), Prof. V. Lakshmikantham, a famous mathematician who founded the only rapidly expanding field of mathematics today, Nonlinear Analysis, founder and editor-in-chief of several scientific journals, and president and founder of the International Federation of Nonlinear Analysts, to tell him that he was a lunatic for associating with me. There were at least five such guys in SciMath. However, in due course they pulled enough rope to hang themselves with academically and are all quiet now. <br /><br />Going back to 1 = 0.99…, it was David Hilbert who recognized almost a century ago that the concepts of individual thought, being inaccessible to others, are ambiguous and cannot be discussed, studied and analyzed collectively. Therefore, they cannot be the subject matter of mathematics. The proper subject matter of mathematics must be objects in the real world, e.g., symbols that we also call concepts that everyone can look at provided they are subject to consistent premises or axioms. Clearly, 1 and 0.99… are distinct objects like apple and orange and to say apple = orange is simply nonsense.<br /><br />As it will turn out SciMath is the best forum in this category in terms of open participation and, naturally, diversity of ideas. The worst in this category, however, is Wikipedia along with its sister website Wikia. Wikipedia requires consensus on posted topic. In other words, it requires uniformity of thought. Wikia specifically bars original research. In effect, they block the progress of science and mathematics which do not thrive on consensus and their progress stands on original research. Between these two extremes the blogs and websites range from good to excellent in terms of diversity of ideas with the only exception of HaloScan and its sister website, DLMSY, which cannot stand contrary opinion. Consequently, they lose bloggers. I identify a few excellent ones in the category of in-betweens: False Proofs, MathForge, WorldPress and Faces of the Moon. I add Knowledgerush in terms of ease in posting – no no username and password which are easy to forget.<br /><br />However, there is a forum that is a class by itself in terms of the quality and level of intellectual discussion: ISCID (International Society for Computing and Intelligent Design (?)). I recommend experts to visit this website.<br /><br />E. E. Escultura<br />Research Professor<br />Lakshmikantham Institute for Advanced Studies and Departments of Mathematics and Physics<br />GVP College of Engineering, JNT University, Visakhapatnam, AP, India<br />E-mail: escultur36@gmail.com * URL: http://users.tpg.com.au/pidro/E. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-5216123557868723972009-09-16T00:34:39.241-07:002009-09-16T00:34:39.241-07:003) Then the exact solutions of Fermat’s equation a...3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation, <br />x^n + y^n = z^n, (F)<br /><br />for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations. <br /><br />4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.<br /><br />E. E. EsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-5700336955101834682009-09-16T00:33:28.878-07:002009-09-16T00:33:28.878-07:00CLARIFICATION ON THE COUNTEREXAMPLES TO FERMAT’S L...CLARIFICATION ON THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM <br />By E. E. Escultura<br /><br />Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:<br /><br />1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2. <br /><br />2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].<br /><br />E. E. EsculturaaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-57033449720976308002009-08-28T22:12:49.502-07:002009-08-28T22:12:49.502-07:006) By reconstructing the defective real number sys...6) By reconstructing the defective real number system into the contradiction-free new real number system and reformulating FLT in the latter, countably infinite counterexamples to it have been constructed showing the theorem false and Wiles wrong.<br /><br />7) In the course of making a critique of the real number system some new results have been found: a) Gauss diagonal method of proving the existence of nondenumerable set only generates a countably infinite set; b) as of this time there does not exist a nondenumerable set; c) only discrete set has cardinality, a continuum has none..<br /><br />8) The new real number system is a continuum, countably infinite, non-Hausdorff and Non-Archimedean and the subset of decimals is also countably infinite but discrete, Hausdorff and Archimedean. The g-norm simplifies computation considerably.<br /><br />E. E. EsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-34494654637781819332009-08-28T22:12:18.680-07:002009-08-28T22:12:18.680-07:004) Vacuous concept generally yields a contradictio...4) Vacuous concept generally yields a contradiction. For example, consider this vacuous concept: the root of the equation x^2 + 1 = 0. That root has been denoted by i = sqrt(-1). The notation itself is a problem since sqrt is a well-defined operation in the real number system that applies only to perfect square. Certainly, -1 is not a perfect square. Mathematicians extended the operation to non-negative numbers. However, the counterexamples to the trichotomy axiom show at the same time that an irrational number cannot be represented by a sequence of rationals. In fact, a theorem in the paper, The new mathematics and physics, Applied Mathematics and Computation, 138(1), 127 – 149, says that the rationals and irrationals are separated, i.e., the union of disjoint open sets.<br />At any rate, if one is not convinced of the mischief that vacuous concept can play, consider this:<br />i .= sqrt(-1) = sqrt1/sqrt(-1) = 1/i = -i or i = 0. 1 = 0, and both the real and complex number systems collapse. <br /><br />5) With respect to Andrew Wiles’ proof of FLT it has two main defects: a) Since FLT is formulated in the inconsistent real number system it is nonsense and, naturally, the proof is also nonsense. The remedy is to first remove the inconsistency of the real number system which I did and reformulate FLT in the consistent number system, the new real number system. b) The use of complex analysis deals another fatal blow to Wiles’ proof. The remedy for complex analysis is in the appendix to the paper, The generalized integral as dual to Schwarz Distribution, in press, Nonlinear Studies. <br /><br />E. E. EsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-6168414316927120782009-08-28T22:10:55.923-07:002009-08-28T22:10:55.923-07:002) The second point is: David Hilbert already knew...2) The second point is: David Hilbert already knew almost a century ago that the concepts of individual thought cannot be the subject matter of mathematics since they are unknown to others and, therefore, cannot be studied collectively, analyzed or axiomatized. Therefore, the subject matter of mathematics must be objects in the real world including symbols that everyone can look at, analyze and study collectively provided they are subject to consistent premises or axioms. Consistency of a mathematical system is important, otherwise, every conclusion drawn from it is contradicted by another. In order words, inconsistency collapses a mathematical system. Consider 1 and 0.99…; they are certainly distinct objects like apple and orange and to write apple = orange is simply nonsense. <br />3) The field axioms of the real number system is inconsistent. Felix Brouwer and myself constructed counterexamples to the trichotomy axiom which means that it is false. Banach-Tarski constructed a contradiction to the axiom of choice, one of the field axioms. One version says that if a soft ball is sliced into suitably little piece and rearranged without distortion they can be reconstituted into a ball the size of Earth. This is a topological contradiction in R^3. <br /><br />E. E. EsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.comtag:blogger.com,1999:blog-12535639.post-40192730539665268982009-08-28T22:08:13.250-07:002009-08-28T22:08:13.250-07:00Needless to say, none of my criticisms of my posit...Needless to say, none of my criticisms of my positions on Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully on this website or across the internet. In peer reviewed publications there is not even a single attempt to refute my positions on these issues. <br /><br />We highlight some of the most contentious issues of the debate. <br />1) Consider the equation 1 = 0.99… that almost everyone accepts. There are a number of defects here. Among the decimals only terminating decimals are well-defined. The rest are ill-defined or ambiguous. In this equation the left side is well-defined as the multiplicative identity element while the right side is ill-defined. The equation, therefore, is nonsense.<br /><br />E. E. EsculturaE. E. Esculturahttps://www.blogger.com/profile/09364110851327981518noreply@blogger.com