A mind-blowing consequence of the MRDP theorem is that there is a multi-variate polynomial which fits on a sheet of paper with the property that the set of values of the first variable which appear in integer solutions are exactly the set of prime numbers.
Any Diophantine equation can be reduced to one of at most 11 variables and degree at most around 10^63. No algorithm can decide solvability in rational numbers for this class of Diophantine equations.
throwaway81523 · 4h ago
That sounds like the coefficients might have to be arbitrarily large. Otherwise all DE's could reduce to a finite set of them, impossible via the MRDP theorem. So it's not so easy to call that bounded complexity.
nine_k · 6h ago
Does this have any practical consequences for cryptography?
ogogmad · 3h ago
Likely not.
badmonster · 9h ago
impressive formalization effort that bridges deep number theory and formal methods
https://en.wikipedia.org/wiki/Formula_for_primes#Formula_bas...
Any Diophantine equation can be reduced to one of at most 11 variables and degree at most around 10^63. No algorithm can decide solvability in rational numbers for this class of Diophantine equations.