Solutions to 4-Color Problem and Goldbach's Conjecture : And also presenting odd algebraic bases for computer science and an attempt to solve wave equation for cylindrical wave

To color a given map we first find its related map with the most mutual adjacency and color it by only four colors, then we trace back. For a large even number there are a large number of pairs of odd numbers sum of the members of each being the even number. We eliminate those pairs that none of the members of each of them is prime and show that number of the remaining pairs is still large. Process of proof shows that there can be no drop to zero in the function of number of the mentioned prime pairs. It is shown that we can take all numbers to odd bases such that we require only about half of the digits required in the current method provided that we introduce negative mark for each digit. Most probably this method will have various applications in computer technology. It is shown that the wave equation cannot be solved for the general spreading of cylindrical wave using the method of separation of variables. But, an equation is presented in case of its solving the above act will have occurred. Also, using this equation, the above-mentioned general spreading of the cylindrical wave for large distances is obtained which contrary to what is believed consists of arbitrary functions.