The word Diophantine is in honor of Diophantus and means integer, so Diophantine equations are ones that have integer solutions. But just because we as human beings specify integer, it does not necessarily mean that the math is limited to them.
For instance x+y = 1, can have x = y = 0.5 as a solution. And simply saying 'Diophantine' does not necessarily force it beyond human choice. So it is fascinating to me when you DO have situations when only integers will do. That is, when the math will not allow anything but integers.
Given x2 - Dy2 = 1, you can get to some really cool equations using elementary methods, valid only for special cases when x = 1 or -1 mod D.
For the x = 1 mod D case:
(D-1)j2 + (j+1)2 = (x+y)2, where j = ((x+Dy)-1)/D
I've shown that
(x+y + j+1) = n2 or 2n2, for some integer n, and:
(x+y - (j+1)) = (D-1)m2 or 2(D-1)m2, for some integer m.
And it is required that j = nm or j = 2nm.
Further either: (n-m)2 - Dm2 = 2, or (n-m)2 - Dm2 = 1
And for the x = -1 mod D case:
(x+y + j-1) = n2 or 2n2, for some integer n.
And (x+y - (j-1)) = (D-1)m2 or 2(D-1)m2, for some integer m.
And again, it is required that j = nm or j = 2nm.
While this time either: (n-m)2 - Dm2 = -2, or (n-m)2 - Dm2 = -1
To understand why those conclusions require integers you have to study the derivation. I think it's fascinating as I noticed a situation where a square is forced by some simple requirements. The trick is that I was able to specify prime p needing to be a factor of D-1, which told the math what I wanted.
For an example with numbers, with D=61, you have some well-known solutions.
x = 1766319049, y = 226153980, so x = 1766319049 = -1 mod 61.
And you get j = 255110030, and can calculate n and m where I'll get one set using positive solutions to the square root (the negative solutions have to work as well):
n = sqrt((x+y+j-1)/2) = 33523,
m = sqrt((x+y-(j-1))/(2(D-1))) = 3805.
It's worth noting that: (n-m)2 - 61m2 = -1, with numbers:
297182 - 61*38052 = -1
This example is great for seeing a case where the math knows that you only want integers! As the rule about squares is only true when you are dealing with integers.
Turns out that human beings have found few such situations, and in many cases it takes human selectivity to only look at integers when the math does not care. But sometimes the math DOES care, and is a brilliant partner in the Diophantine search.
As a partner the math has infinite intelligence, which is something we human beings cannot, as finite creatures, completely understand. But the math doesn't care about that either.
It's actually kind of freaky. And a mental exercise is trying to find an error in my conclusion that only integers are allowed. Then wonder: how does the math do it?
And contemplate that the math covers infinity, which can make you feel really appropriately small.
So to get integers only above the math has to consider an infinite number of results!
No human can compete in that arena.