Guest wrote:FYI: "Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers." --

https://en.m.wikipedia.org/wiki/Hilbert%27s_tenth_problemMy tentative view is that we, humans, are not yet clever enough to devise a general algorithm to solve Hilbert's Tenth Problem. There is a current 'proof' of the Matiyasevich's Theorem which states the problem is insolvable. We must challenge the validity of this grand and important result. The 'proof' may be seriously flawed...

Dave,

https://www.researchgate.net/profile/David_Cole29/amp

"Matiyasevich's Theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says:

Every computably enumerable set is Diophantine.

A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0.

Conversely, every Diophantine set is computably enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0. Now we make an algorithm which simply tries all possible values for n, x1, ..., xk (in, say, some simple order consistent with the increasing order of the sum of their absolute values), and prints n every time f(n, x1, ..., xk) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x1, ..., xk) = 0 has a solution in x1, ..., xk.

Proof technique:

Yuri Matiyasevich utilized a method involving Fibonacci numbers, which grow exponentially, in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam – hence, MRDP – had shown that this suffices to show that every computably enumerable set is Diophantine."