Conclusion to introduction of Number Theory : Number Theory #number #theory

 Conclusion  to introduction of  Number Theorem :

                   ( part 5 continuation )

                     We conclude this introduction with a brief mention of some outstanding unsolved problems concerning prime numbers.

1           Goldbach’s problem : Is there an even number >2 which is not the sum of two                                                      primes?

            *  Is there an even number >2 which is not the difference  of two primes

3.         * Are there infinitely many twin primes?

4.         * Are there infinitely many Mersenne primes , that is, primes of the form 2^p – 1                where p is prime?

5.         * Are there infinitely many composite Mersenne numbers?

6.         * Are there infinitely many Fermat primes, that is, primes of the form 2^{2^n} +1?

7.         * Are there infinitely many composite Fermat numbers?

8.         * Are there infinitely many primes of the form x²+1, where x is an integer?

9.         * Are there infinitely many primes of the form x² + k ?

10        * Does there always exist  at least one prime between n² and (n+1)² for every                      integer n ≥ 1?

11        * Does there always exist at least one prime between n² and n² + n for every                        integer n>1?

12       * Are there infinitely many primes whose digits are all ones?

                    The professional mathematician is attracted to number theory because of the way all the weapons of modern mathematics can be brought to bear on its problems. As a matter of fact, many important branches of mathematics had their origin in number theory. For example, the early attempts to prove the prime  number theorem stimulated the development of the theory of functions of a complex variable, especially the theory of entire functions. Attempts to prove that the Diophantine equation xⁿ + yⁿ = zⁿ has no nontrivial solution if n ≥ 3 led to the development of algebraic number theory, one of the most active areas of modern mathematical research. Even though Fermat’s conjecture is still undecided, this seems unimportant by comparison to the vast amount of valuable mathematics that has been created as a result of work on this conjecture. Another example is the theory of partitions which has been an important factor in the development of combinatorial analysis and in the study  of modular functions.

                    There are hundreds of unsolved problems in number theory. New problems arise more rapidly than the old ones are solved, and many of the old ones have remained unsolved for centuries. As the mathematician Serpinski once said, “ the progress of our knowledge of numbers is advanced not only by what we already know about them,  but also by  realizing what we yet do not know about them. “

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