buck9

Comment Threads

• Trottelreiner commented on On hold
  As for the mathematical meaning, it changes quite a bit, since a diophantine equation implies only integers, e.g. if it's not "completely rational integer coefficients", it's not a diophantine equation. So, Hilbert mentioning these conditions is all somewhat redundant, as mentioned. "Elaborated code", as I mentioned......
• Trottelreiner commented on On hold
  For a quite famous diophantine equation, take Fermat's theorem....
• Elderly Cynic commented on On hold
  Not entirely. The definition occurs in the two equivalent forms, one with integer coefficients and one with integer rational ones. Note that you can have rationals based on polynomials and other rings, as well as integers. If I recall, the term Diophantine wasn't as universally used in the 19th century as it was by the mid-20th. But I may well be misremembering....
• Trottelreiner commented on On hold
  Hm, the definition of "rational" sounds somewhat strange to me, I only know it as "quotient of two integers". But apparently there is a general definition of "Diophantine" as "somewhat restricted": https://en.wikipedia.org/wiki/Polynomial_Diophantine_equation Though there are no citations on this article, and I have somewhat dropped out with the mathematicians in my circle of friends, so I can't verify it. As for my books, still in storage......
• Elderly Cynic commented on On hold
  See, for example: https://math.dartmouth.edu/opencalc2/cole/lecture19.pdf Part of the reason that such things are interesting is that, if A is a ring, the ratio of two elements in A is a field. I am VERY rusty, so may have forgotten whether there are any other conditions - I could still work that out, but not now....