Diophantine Equations

In Diophantine problems the number of equations is less than thenumberofvariables(i.e.theequationsareundecidable).Thecoefficientsof the variables in these equations are integers and such integer valuesof the variables have to be found which satisfy these equations. That'swhy all Diophantine Equations havecoefficientsofthevariablesasintegers.

 

Example-

x*y = 4

 

Usually, this equation cannot be solved as it has infinite solutions. However, if we consider x and y to be positive integers only, then we realize that x and y are just factors of 4 which multiply to give 4. Now, the finite solutions can be found easily.

x,y can be (1,4) or (2,2) or (4,1).

 

Due to only integer answers, Diophantine equations are used widely in the world. For example, in chemistry, especially in balancing equations (for chemical reactions). In chemical reactions, it is possible only for whole atoms or moles to react. Half atoms cannot react so these equations can help chemists balance an equation easily. Another example in chemistry itself would be finding out a compound. Example, if a compound weighs some m units and consists only of elements A, B and C with atomic masses x, y and z, we form the equation:

p*x + q*y + r*x = m

Where p, q and r would denote number of atoms of elements A, B and C.

 

These equations can expand into other fields as well, such as the real-life application of geometry. For example, while building complex structures such as bridges or rockets, even a small error can prove to be disastrous. So having lengths like irrational square roots which cannot be precise physically results in expensive inaccuracies. Instead, solutions to the given equations for structures like these as integers can be very beneficial.