Nikos Mantzakouras | National & Kapodistrian University of Athens (original) (raw)

Papers by Nikos Mantzakouras

The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics; a sim... more The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics; a simple relation of square numbers that encapsulates the glory of mathematical science. It is also justifiably the most popular yet sublime theorem in mathematical science. The starting point was Diophantus' 20th problem (Book VI of Diophantus' Arithmetica), which for Fermat is for n = 4 and consists in the question whether there are right triangles whose sides can be counted as integers and whose surface can be square. The generalized method we use here is based on elementary inequalities and gives a solution to any Diophantine equation of degree n with respect to the number of variables d. It is a method that does not exclude other equivalent analytical methods for finding specific solutions. However, the method proposed generalizes to any exponent n with number of variables d. An explicit proof is also given for variables in both parts with equal or different number of variables, also of degree n. As a direct application it is Fermat's Last Theorem and of course has much broader generalization to symmetric or asymmetric Diophantine equations.

May, 2024

Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a famous list compiled... more Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a famous list
compiled in 1900 by David Hilbert.It entails the assumption existence of the solution
for all 7th degree equations using algebraic (variant: continuous) functions of two
arguments. It was first presented in the context of nomography, and in particular
"nomographic construction" a process whereby a function of several variables is
constructed using functions of two variables. The variant for continuous functions was
resolved affirmatively in 1957 by Vladimir Arnold when he proved the Kolmogorov–
Arnold representation theorem, but the variant for algebraic functions remains
unresolved. Hilbert's continuing fascination with the 13th problem is evident from the
fact that in his last mathematical paper [2,9,10], published in 1927, in which he
reported the status of his problems, Hilbert devoted 5 pages to the 13th problem and
only 3 pages to the other 22 problems.

Analyzing the paradoxes with the theory of relativity, we stop at the Doppler effect Paradox, whi... more Analyzing the paradoxes with the theory of relativity, we stop at the Doppler effect Paradox, which hides a very important secret, that the relation of time and space is with the Lorentz transformations wrong for the SS' system and Multiple moving Frames. On the contrary, the relations for the laboratory are correct, i.e. those related to the Mickelson experiment. In this study, we prove that the relations of time and space and therefore of the following quantities, velocity and acceleration, are different from those given by Lorentz. The relation connecting energy and mass with velocity is also wrong and the correct relation is completely different. The speed of light in accordance with the theories of spherical electromagnetic wave and the FRENSEL relation proves that it is probably not constant but is influenced by the density of space and spacetime in general. For the twin paradox, we will not mention it because the changes in times and lengths are observational, not real. Therefore, for the paradox of Gemini, it is obvious that the time in essence for the life of the moving person does not change, just the clock is delayed and the length seems shorter that it travels. Just changes that are electronic, mechanical or generally observational and not real. Also, a generalization is given for fractional calculus regarding the expansion of time according to fractional integration.

$Research paper thumbnail of Approximate methods for solving differential fractional equations$

The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics; a sim... more The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics; a simple relation of square numbers that encapsulates the glory of mathematical science. It is also justifiably the most popular yet sublime theorem in mathematical science. The starting point was Diophantus' 20th problem (Book VI of Diophantus' Arithmetica), which for Fermat is for n = 4 and consists in the question whether there are right triangles whose sides can be counted as integers and whose surface can be square. The generalized method we use here is based on elementary inequalities and gives a solution to any diophantine equation of degree n with respect to the number of variables d. It is a method that does not exclude other equivalent analytical methods for finding specific solutions. However, the method proposed generalizes to any exponent n with number of variables d. An explicit proof is also given for variables in both parts with equal or different number of variables, also of degree n. As a direct application it is Fermat's last theorem and of course has much broader generalization to symmetric or asymmetric diophantine equations. Part I. Generalized theorem of Diophantine equation of degree n with d number of variables

In this paper, we explain a new Iterative Method-Fixed Point and develop its convergence theory f... more In this paper, we explain a new Iterative Method-Fixed Point and develop its convergence theory for finding approximate solutions of nonlinear equations in the setting of Banach spaces. First, we discuss the convergence analysis of our method by separating the equation into functions from which it is derived and the remaining part of the equation, according to the Generalized Theorem[1] for solving polynomial and transcendental equations. Solving an equation, either polynomial or transcendental, is solved only by categorizing the roots and not by some procedure of searching in individually intervals and this way was followed in the past by all existing methods. But the methods developed in the past are limited to isolated intervals without general acceptance. Finally, the method is strengthened by Newton's method to speed up the finding of the roots. At the end of the analysis we give several examples with application of the method

Solving nonlinear equations in any Banach space (real or complex nonlinear equations, nonlin-ear ... more Solving nonlinear equations in any Banach space (real or complex nonlinear equations, nonlin-ear systems, and nonlinear matrix equations, among others) is a non-trivial task involving many areas of science and technology. Usually the solution is not readily accessible and requires an approach that uses iterative algorithms. In the present work, which is complementary to previously published work, the focus is mainly on the design, convergence analysis and stability of new iterative schemes for solving nonlinear problems and their application to scientifically important problems. The methods we will use are two. One involves the well-known Newton with Jacobi-matrix method, but with the help of the generalized solution theorem [1] & [3]. The second, that is new, deals with solving equations using the GRIM method, which is an iterative method and which is again based on the generalized theorem solve equations [1] & [3]. The potential of the generalized theorem is that it gives us the maximum number of real or complex roots by a cate-gorization procedure based on integer coefficients in general, rather than a stepwise or random search in some the interval.

Mantzakouras Nikos :DOI :10.13140/RG.2.2.24591.92329/14, 2023

The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics. A sim... more The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics. A simple relation of square numbers, which encapsulates all the glory of mathematical science, is also justifiably the most popular yet sublime theorem in mathematical science. The starting point was Diophantus' 20 th problem (Book VI of Diophantus' Arithmetica), which for Fermat is for n = 4 and consists in the question whether there are right triangles whose sides can be measured as integers and whose surface can be square. This problem was solved negatively by Fermat in the 17 th century, who used the wonderful method (ipse dixit Fermat) of infinite descent. The difficulty of solving Fermat's equation was first circumvented by Willes and R. Taylor in late 1994 ([1],[2],[3],[4]) and published in Taylor and Willes (1995) and Willes (1995). We present the proof of Fermat's last theorem and other accompanying theorems in 4 different independent ways. For each of the methods we consider, we use the Pythagorean theorem as a basic principle and also the fact that the proof of the first degree Pythagorean triad is absolutely elementary and useful. The proof of Fermat's last theorem marks the end of a mathematical era; however, the urgent need for a more educational proof seems to be necessary for undergraduates and students in general. Euler's method and Willes' proof is still a method that does not exclude other equivalent methods. The principle, of course, is the Pythagorean theorem and the Pythagorean triads, which form the basis of all proofs and are also the main way of proving the Pythagorean theorem in an understandable way. Other forms of proofs we will do will show the dependence of the variables on each other. For a proof of Fermat's theorem without the dependence of the variables cannot be correct and will therefore give undefined and inconclusive results.

Mantzakouras Nikos :DOI:10.13140/RG.2.2.10888.34563/7, 2023

Riemann's zeta function is one of the most important and fascinating functions of Euler's mathema... more Riemann's zeta function is one of the most important and fascinating functions of Euler's mathematics. By analyzing the material of Riemann's hypothesis, we divide our analysis into the function ζ(z) and the proof of the conjecture, which has very important consequences in terms of the distribution of prime numbers. The proof of Riemann's hypothesis follows from the simple logic,that when two properties are related, i.e. these equations are zero i.e. ζ(z) = ζ(1 − z) = 0 while they have the proven 1 − 1 property of Riemann's function ζ(z), then the Hypothesis will be true, because it has been proved from before simultaneously that Riemann's functional equations hold. Therefore, there will be apply ζ(z) = 1/2 {because ζ(z) = ζ(1 − z) = 0 and also because ζ(z) as well as ζ(1 − z) are 1-1}. This, being true, will give the direction of all non-trivial roots to all be on the critical line, with a value of 1/2 on the real axis. Furthermore, to strengthen the proof we consider "5 important requirements" that clearly prove that 100% of the roots of ζ(z) = 0 are only on the critical line. There is of course the case ζ(q * z) = 0 with q > 1 in R,, which has different critical lines, but all those who deal with the Hypothesis of Riemann, do not mention it , because they claim that this form was not defined by Riemann from the begining. But if we generalize the hypothesis to ζ(q * z) = 0, q > 1 in R then we have infinite critical lines that originate from and are related to the baseline critical line of Re(z) = 1/2, but are not identical to it and lie within the interval (0, 1). In summary and by analytically examining and other similar functions to ζ(z), we get zeros with Re(z) in the interval (0, 1) for the cases ζ(q * z) = 0 where q > 1 in R that mentioned, for Generalized case G − ζ(z, q) = 0 with 3 general cases out of 5 which is the generalized sum including ζ(z) and end for the Davenport Heilbronn case, which has zeros of a periodic Dirichlet series with different critical lines, within the interval (0, 1). We are likely to have other constructed cases with relevance of ζ(z). Therefore, whether or not the Riemann Hypothesis, holds depends on whether or not we accept the fact that the Riemann Hypothesis is or is not fully mentioned, to functional equation ζ(z) = 0.

Mantzakouras Nikos (DOI: 10.13140 / RG.2.2.18719.89760), 2019

If we simulate the 2 equations, it turns out that when n> 2 for c^n + b^n = a^n if n = 2k + 1 , k... more If we simulate the 2 equations, it turns out that when n> 2 for c^n + b^n = a^n if n = 2k + 1 , k> = 1 in Z +, only 2 things can happen 1. (sinx = 0 and cosx = 1) or ( sinx = 1 and cosx = 0) 2nd. if n = 2k, (cosx = + / - 1 and sinx = 0) or ( sinx = + / - 1 and сosx = 0) will hold, where k> 1 in Z +. Otherwise (sinx) ^ n + (cosx) ^ n <1 (n = 2k or n = 2k + 1) and therefore equality is impossible .Only in the case n = 2 is equal to 1 as a maximum i.e. (sinx) ^ n + (cosx) ^ n = 1. On the logic that if the limiting case = 1 arises then the results found in the paper are obtained.Οtherwise obviously we will have a form of inequality so the equality will not hold and hence the solution for n>2 is falsified..This is a simulation method since we assume that the angle for exponent n>2 is approximately equal to exponent n=2..Τhis paper is a supplement to the 2nd paper found on the website,https://www.academia.edu/39519597/FERMAT_LAST_THEOREM_PROOF

Mantzakouras Nikos

If Fermat’s Last Theorem were false, this would require either a conspiracy theory , or a quasi-c... more If Fermat’s Last Theorem were false, this would require either a conspiracy theory , or a quasi-conspiracy theory.

“The conspiracy theory, of course, would be that mathematicians as a body know that Fermat’s Last Theorem is false, but do not want everyone else to know this, so they claim that they have verified the proof and found it valid, while in reality there are flaws in it and they know about them.

The quasi-conspiracy theory would be that mathematicians as a body believe that Fermat’s Last Theorem is true, but that they consistently fail in their attempt to verify the proof. There is a mistake in it, but each time someone tries to verify it, they fail to notice the mistake.

The reason to call this a quasi-conspiracy theory is that the most reasonable way for this to happen is if mathematicians as a body have motivations similar to the mathematicians in the case of the actual conspiracy, motivations that cause them to behave in much the same ways in practice.

We can see this by considering a case where you would have an actual conspiracy. Suppose a seven year old child is told by his parents that Santa Claus is the one who brings presents on Christmas Eve. The child believes them. When he speaks with his playmates, they tell him the same thing. If he notices something odd, his parents explain it away. He asks other adults about it, and they say the same thing.

The adults as a body are deceiving the child about the fact that Santa Claus does not exist, and they are doing this by means of an actual conspiracy. They know there is no Santa Claus, but they are working together to ensure that the child believes that there is one.

What is necessary for this to happen? It is necessary that the adults have a motive quite remote from truth for wishing the child to believe that there is a Santa Claus, and it is on account of this motive that they engage in the conspiracy.

In a similar way, suppose that mathematicians as a body were deluded about Fermat’s Last Theorem. Since they are actually deluded, there is no actual conspiracy. But how did this happen? Why do they all make mistakes when they try to verify the theorem? In principle it might simply be that the question is very hard, and there is a mistake that is extremely difficult to notice. And in reality, this may be the only likely way for this to happen in the case of mathematics. But in other cases, there may be a more plausible mechanism to generate consistent mistakes, and this is wishful thinking of one kind or another. If mathematicians as a body want Fermat’s Last Theorem to be true and to be a settled question, they may carelessly overlook mistakes in the proof, in order to say that it is true. Technically they are not making a deliberate mistake. But in practice it is the lack of care about truth, and the interest in something opposed to truth, which makes them act as a body to deceive others, just as an actual conspiracy does.”

The fact is, of course, that the proof of the Wiles is not widely accepted and understood, therefore it is a proof of conspiracy .

However, the Last theorem of fermat as a theorem is correct, the healthy mathematical community expects a proof accessible at its average spiritual level so that it is a clear proof and within the real mathematical theories.

The purpose of this paper is to provide a proof that is simple , understandableand and widely accepted. Any critique is acceptable. Thanks for the scientific world of academia.edu.

Mantzakouras Nikos (DOI: 10.13140/RG.2.2.10888.34563)

The Riemann zeta function is one of the most Euler's important and fascinating functions in mathe... more The Riemann zeta function is one of the most Euler's important and fascinating functions in mathematics. By analyzing the material of Riemann's conjecture, we divide our analysis in the ζ(z) function and in the proof of the conjecture, which has very important consequences on the distribution of prime numbers. The proof of the Hypothesis of Riemann result from the simple logic, that when two properties are associated, (the resulting equations that based in two Functional equations of Riemann), if zero these equations, ie ζ(z) = ζ (1-z)= 0 and simultaneously they to have the proved property 1-1 of the Riemann function ζ(z), the hypothesis will be true.Thus, there is not margin for to non apply that the Re (z) = 1/2 {because ζ (z) = ζ (1-z)=0 and also ζ(z) as and ζ(1-z) are 1-1}.This, after it stands, will gives the direction of all the non-trivial roots to be all in on the critical line, with a value in the real axis equal 1/2. Part I. Proof Riemann Hypothesis from Theorems 1&3. The R-Hypothesis focuses on the point where we must prove that if s = Re(s) + Im(s)*I … i) The functions ζ(s) and ζ(1-s) are 1-1 on the critical strip. ii) The common roots of the equations ζ(s)-ζ(1-s) = 0 they have Re(s)=Re(1-s)=1/2 within the interval (0,1) and determine unique position, which is called critical line. In Part II, we examine the distribution of prime numbers. Thus, if we ask to prove whether the hypothesis is correct, we must ask for the values of the critical lines that confirm the Riemann hypothesis. But, the Riemann hypothesis does not hold everywhere in the interval (0,1).But we have critical lines and inside of intervals (-1,0) or (1,+∞) or (-∞,-1) specifically in the cases of the general equation ζ(q*z)=0,q>1 or q<-1 or (0<q<1) or (-1<q<0)& q∊R. But there is 1 solution of critical line the 1/2 (q=1) and 1solution of critical line the -1/2(q=-1) and for non-trivial roots only if ζ(s) =0 or ζ(-s)=0 because is 1-1 and this is also proved in functional equations as we showed in the paper. Here we consider a case of the functional equations ζ(z)=0 given by Riemann himself.But there is a generalization to this. Also, this is another Hypothesis,which needs a much deeper analysis. There is a contradiction from a 2007 paper by Davenport Heilbronn, that there are other points z(i) for which ζ(z(i))=0 that do not belong to the critical line. This is wrong (because here we have ζ(z)=0) both as an assumption and as a value of ζ(z(i)), simply because it is not zero. It goes without saying that this story has set back the proof and there are still other mathematicians and Russians who have relied on this paradox in their opinion and it has delayed or confused the mathematical community. Also these points do not seem to correspond to a critical line, they are scattered and obviously do not refer to the case as defined by Riemann.Finally, the cases ζ(z,q)=0 or ζ(q,z)=0 ,q∊R, should be considered more generally to clarify whether there are zeros and where they exist and if they correspond to critical lines.Therefore, if apply or not the Riemann hypothesis holds depends on whether or not we accept that we are considering only ζ(z)=0. If in the definition of the Hypothesis the other forms or part of them also hold , i.e. ζ(q*z)=0 or ζ(q,z)=0 or ζ(z,q)=0 with q∊R in general, then the Hypothesis does not hold in general, but only in the special case but only in the special case which was mentioned.

Mantzakouras Nikos

In International Conference From Nina Ringo on Mathematics and Mechanics 1-2 June 2017, Nice, Fra... more In International Conference From Nina Ringo on Mathematics and Mechanics 1-2 June 2017, Nice, France
Dedicated to the memory of David Hilbert (Jan uary 23, 1862 – February 14, 1943)

Mantzakouras Nikos

Distribution and finding intervals contained of the zeros of the Riemann Zeta function. Abstract.... more Distribution and finding intervals contained of the zeros of the Riemann Zeta function. Abstract.. One of the most celebrated problems of mathematics is the Riemann hypothesis which states that all the non trivial zeros of the Zeta-function lie on the critical line s = 1/2. all the non trivial zeros lie in the critical strip, the number of such zeros with ordinate less than T is proportional to T log T, most zeros concentrate along the critical line s = 1/2, there exists an infinity of zeros on the critical line and moreover, more than two fifth of the zeros are on the critical line.Finding the roots by Lagrange method demarcating the intervals whose are not trivial zeros of the Zeta-function.

Mantzakouras Nikos

from book Solve equations Use mathematica!!!(In greece)

Mantzakouras Nikos

The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions ... more The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.

May, 2024

$Research paper thumbnail of Approximate methods for solving differential fractional equations$

The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics; a sim... more The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics; a simple relation of square numbers that encapsulates the glory of mathematical science. It is also justifiably the most popular yet sublime theorem in mathematical science. The starting point was Diophantus' 20th problem (Book VI of Diophantus' Arithmetica), which for Fermat is for n = 4 and consists in the question whether there are right triangles whose sides can be counted as integers and whose surface can be square. The generalized method we use here is based on elementary inequalities and gives a solution to any diophantine equation of degree n with respect to the number of variables d. It is a method that does not exclude other equivalent analytical methods for finding specific solutions. However, the method proposed generalizes to any exponent n with number of variables d. An explicit proof is also given for variables in both parts with equal or different number of variables, also of degree n. As a direct application it is Fermat's last theorem and of course has much broader generalization to symmetric or asymmetric diophantine equations. Part I. Generalized theorem of Diophantine equation of degree n with d number of variables

Mantzakouras Nikos :DOI :10.13140/RG.2.2.24591.92329/14, 2023

Mantzakouras Nikos :DOI:10.13140/RG.2.2.10888.34563/7, 2023

Mantzakouras Nikos (DOI: 10.13140 / RG.2.2.18719.89760), 2019

Mantzakouras Nikos

The fact is, of course, that the proof of the Wiles is not widely accepted and understood, therefore it is a proof of conspiracy .

The purpose of this paper is to provide a proof that is simple , understandableand and widely accepted. Any critique is acceptable. Thanks for the scientific world of academia.edu.

Mantzakouras Nikos (DOI: 10.13140/RG.2.2.10888.34563)

Mantzakouras Nikos

from book Solve equations Use mathematica!!!(In greece)

Mantzakouras Nikos

Finding a definite limit, for a proof of Fermat's last theorem

A answere simple in Integers is x=2 …Because (3,4,5) is Pythagorean triad.But we also want comple... more A answere simple in Integers is x=2 …Because (3,4,5) is Pythagorean triad.But we also want complex roots and everything wants proof. We try Within interval Abs[x]<100 and we are looking for the roots…For the first we divide with the highest integer and we have id est…5 =>the equation take the form… (3/5)^x+(4/5)^x=1. This equation is the final .This we will try to solve!!! Αccording to the method of the solved equation,,G.R.L(Generalized Resol.Theorem of Lagrange, https://www.academia.edu/8218891/SOLVE_EQUATIONS)