Please find related article in my page, considering Fermat's last theorem according to trigonometry & polar coordinates. I appreciate your comments. Thank you.
It proves just that the sides of right-angled triangles can't be solutions to FLT if n>2. Try to express, for example, x=3; y=3 and z=10000 as functions of single angle phi and length r
In the article "Modified Pythagoras and Fermat Formula (MPFF)", what do you want to express by drawing two triangles, for in right-angled triangles the written properties between angles and sides do not hold.
How do you verify that, for example, (5.4513617784964..)^3+ (3.7797631496846..)^3 = 216. Try comparing, for example, 355/113 and pi.
In the article I didn't find a place where variables "x-side" and "y-side" would be explained.
I don't get why you had to go though all the trouble of using trigonometry, if a simple formula y=(r^n-x^n)^(1/n) for chosen r,n and x does the trick. Formula for finding r is easy as well: r=(x^n+y^n)^(1/n), given x, y and n. A bit more interesting would be formula for finding the exponent n, if r, x and y are given. I think LambertW function might be involved.
In pictures involving Graphing Calculator, it would be helpful, if you could add what are you trying to find in each input line.
Fermat's eq and also Beal's conjecture can be considered as MPFF. Simply using polar coordinates both Fermat & Beal can be calculated as I have shown also showing above n is two as natural number therefore according to trigonometry there must be at least one or more irrational number in the related equations.
In the article "Modified Pythagoras and Fermat Formula (MPFF) and Beal’s Conjecture", I still don't see what benefit gives the trigonometry to FLT or Beal's conjecture, except for giving some kind of geometrical interpretation in non-Euclid space. Let me give you an example of useful application. There are few problems involving finding all triangles, in which length of sides, as well some element of triangle(for example, median) are rational numbers. There are known formulas for calculating the length of median. That's why trivial approach would be : loop through the lengths of sides, and check whether the length of median is rational. However, the usage of theory of elliptic curves simplifies the task. Write equation of corresponding elliptic curve, find a generator, and calculate the multiples of generator. It's obvious which is less daunting task.
Again, numerical verification with decimal representation of irrational numbers might prove something in medicine but not mathematics.
For atan calculations, why do you need to use square root of the division. Moreover, why use atan if using acos for calculation of a, and asin for calculation of b might simplify the formulas. I don't know whether will it oversimplify.
On page 8, it's obvious that r^n*cos(theta)*cos(theta) and r^n*sin(theta)*sin(theta) are both natural numbers. However, cos(theta) and sin(theta) are integers only in few cases. That's why it's possible that cos(theta) will remove all common factors of r^n, r^n*cos(theta)*cos(theta) and r^n*sin(theta)*sin(theta). Therefore it does not prove that GCD won't be 1.
Dear Valdemārs Plociņš
Did you look at my last article " Probably Also Fermat’s Own Proof for FLT (Fermat’s Last Theorem) " where I simplified right angle triangle as sides being ( hypotenuse=1, horizontal= c and vertical= (1-c^2)^(1/2), with this formula, again I reach correct values by axiom of irrationality. You also ask me to prove what ı claim as axiom. Since such a conclusion is reality how can any mathematician can prove this really I don't know. I explained this to all world as my axiom of irrationality, if there is anybody who can prove, I congratulate him or her. Also you have to admit that I am not a professional mathematician as also Fermat was, so nobody can expect from me formal a
(Continue my above script) So nobody can expect from me a type of formal abstract & complex mathematical proof of professional/academic mathematicians. My MPFF, right angle triangular logic along with my new axiom of irrationality works quite well for FLT and please remember the conjecture as " no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two." Clearly my papers explain why it is so. thank you for your very fine comments.
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