On a Possible Generalization of Fermats Last Theorem

In this short paper we propose four conjectures in synthetic geometry that generalize Erdos-Mordell Theorem, and three conjectures in number theory that generalize Fermat Numbers.

In this paper we present an elementary proof for a special case of Fermat's last theorem for specific category of a, b and c. In fact, we assume that $n$ is prime and $4\rvert (n+1),$ then for $a,b$ and $c$ that $ n\nmid abc$ the equation $a^{2n} = b^{2n} + c^{2n}$ does not have any solution in natural numbers.

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than c. Triples of numbers satisfying a+b=c are called abc triples if the product of their distinct prime divisors is strictly less than c. We catalog what is known about ...

This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterl\'e and the author's diophantine conjecture for algebraic points of bounded degree. It also shows that the new conjecture is implied by the earlier conjecture. As with most of the author's conjectures, this new conjecture stems from analogies with Nevanlinna theory; in this case it corresponds to a Second Main Theorem in Nevanlinna theory with truncated counting functions. The original a...

This paper has been withdrawn by the author for further investigation.

This paper describes a method used to construct infinitely many probable counterexamples of the abc conjecture over the rational integers.

In the present article, we extend previous results of the author and we show that when $K$ is any quadratic imaginary field of class number one, Fermat's equation $a^p+b^p+c^p=0$ does not have integral coprime solutions $a,b,c \in K \setminus \{ 0 \}$ such that $2 \mid abc$ and $p \geq 19$ is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.

In this short article we do not prove Fermat's last theorem. We show that the number 2 is an exceptional number in this theorem.

We revisit a subexponential bound for the $abc$ conjecture due to the first author, and we establish a variation of it using linear forms in logarithms. As an application, we prove an unconditional subexponential bound towards the $4$-terms $abc$ conjecture under a suitable hypothesis on the size of the variables.

In this paper, we generalized the classical Fermat point, proved the sufficient and necessary condition for uniqueness and existence for the generalized Fermat point(GFP) theorem, and discuss some interesting geometric property of the generalized Fermat point.