We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their ...We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their origin then we classified them into two categories according to their classes, we showed in which context two prime numbers which differ from 6 are called sexy and in what context they are said real sexy prime. For those whose difference is equal to 4, we showed their origin then we showed that two prime numbers which differ from 4, that is to say two cousin prime numbers, are successive. We made an observation on the supposed prime numbers then we established two pairs of equations from this observation and deduced the origin of the Mersenne number and that of the Fermat number.展开更多
Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number great...Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A<sup>2</sup> + B<sup>2</sup> + C<sup>2</sup> + D<sup>2</sup> + so on =A<sub>n</sub><sup>2 </sup>where all are natural numbers.展开更多
The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋...The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.展开更多
文摘We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their origin then we classified them into two categories according to their classes, we showed in which context two prime numbers which differ from 6 are called sexy and in what context they are said real sexy prime. For those whose difference is equal to 4, we showed their origin then we showed that two prime numbers which differ from 4, that is to say two cousin prime numbers, are successive. We made an observation on the supposed prime numbers then we established two pairs of equations from this observation and deduced the origin of the Mersenne number and that of the Fermat number.
文摘Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A<sup>2</sup> + B<sup>2</sup> + C<sup>2</sup> + D<sup>2</sup> + so on =A<sub>n</sub><sup>2 </sup>where all are natural numbers.
文摘The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.
