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4.3 Diophantine Equations

Consider the following Diophantine equation (the polynomial equations, in any number of variables, for which all the coefficients and all the solutions must be integers [13]):

6w + 2x2-y3 =0

5xy - z2 + 6 = 0

w -w+2x-y=z-4=0

When solved, the solution comes out to be: w = 1; x = 1; y =2; z= 4;

Now consider another Diophantine equation:

6w + 2x2-y3 = 0

5xy-z2+6 = 0

w -w+2x-y-z-3=0

This equation has no solution (because by its first equation, y must be even, by second equation z must be even also but this contradicts its third equation: whatever w is, because w2 - w is  always even, while 3 is an odd number).

The first equation can be solved by writing a computer algorithm, which just slavishly tries all sets of integers one after another, and after a certain number of trials, it finds: w= l;x= l;y=2;z =  4isthe solution. When the second equation is given to the same algorithm, it will never halt for it will not get any combination of numbers which solve the equation.

While human being using his conscious brain can easily tell whether such an equation has solution or not, no computer algorithmusing artificial intelligence can do this.

While human being using his conscious brain can easily tell whether such an equation has solution or not, no computer algorithm     using     artificial intelligence can do  this.   If consciousness       could      be simulated   on   a   computer, then   there    must    exist    a computational algorithm for the above mathematical problem, which could tell in yes/no that such numbers can exist or  not.Russian mathematician Yuri Matiyasevich [14] demonstrated that there can't be any computer algorithm which decides yes/no systematically to the question of whether a system of  Diophantine equation has a solution.