devotion & mathematics

                                                                                                    Contents :                                                                                  1. Vector function of a scalar variable          2. Limit of a vector function          3. Continuity of a vector function          4. Derivative of a vector function          5. Higher order derivatives          6. Propert...

  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, p...

    Historical Introduction to Number Theory : Part -5                             ( part 4 continuation ) In 1851 the Russian mathematician Chebyshev made an important step forward by proving that if the ratio 𝛑 (x  ) log x / x  did tend to a limit , then this limit must be 1. However he was unable to prove that the ratio does tend to a limit.  In 1859 Riemann attacked the problem with analytic methods, using a formula discovered by Euler in  1737 which relates the prime numbers to the function          ζ (s)=   Σ 1/n s  for a real s >1. Riemann consider complex values of small s and outlined an ingenious method for connecting the distribution of primes to properties of the function ζ (s). The mathematics needed to justify all the details of his method had not been fully developed and Riemann was unable to  c...

           Historical Introduction to Number Theory : Part -4                                                                   ( part 3 Continuation )      On the other hand , the tables indicate that consecutive primes, such as 3 and 5 or 101 and 103 keep recurring. Such pairs of primes which differ only by 2 are known as twin primes . There are over 1000 such pairs below 100,000 and 8000 below 1,000,000. The largest pair known to date is 76 . 3 139 -1 and   76 . 3 139 +1. Many mathematicians think there are infinitely many such pairs, but no one has been able to prove this as yet.  One of the reasons for this irregularity in distribution of primes is that no simple formula exists for producing all the primes.  Some formulae do yield many primes. For exa...

   Diophantine Equation :         A Diophantine equation is a polynomial equation in one or more variables where only integer solutions are sought . General form: f ( x 1 , x 2 , … , x n ) = 0 f(x_1, x_2, \dots, x_n) = 0 f ( x 1 ​ , x 2 ​ , … , x n ​ ) = 0 where: f f f has integer coefficients Solutions must be integers (sometimes non-negative integers) 📌 Named after Diophantus of Alexandria (3rd century). 2. Types of Diophantine Equations 2.1 Linear Diophantine Equations Form: a x + b y = c ax + by = c a x + b y = c where a , b , c ∈ Z a, b, c \in \mathbb{Z} a , b , c ∈ Z Condition for solutions: gcd ⁡ ( a , b ) ∣ c \gcd(a, b) \mid c g cd ( a , b ) ∣ c General solution: If d = gcd ⁡ ( a , b ) d = \gcd(a, b) d = g cd ( a , b ) , and ( x 0 , y 0 ) (x_0, y_0) ( x 0 ​ , y 0 ​ ) is one solution, then: x = x 0 + b d t , y = y 0 − a d t , t ∈ Z x = x_0 + \frac{b}{d}t,\quad y = y_0 - \frac{a}{d}t,\quad t \in \mathbb{Z} x = x 0 ​...