I found a lot of other much stronger restrictions that practically completely restrict one of the 3 variables as well as put a bunch of restrictions on the other two and I think it can be done. Or maybe not idk it's still huge.
Like I saw in a comment you said that you got that c < k(a+b) for some k and similar bounds for a and b. This doesn't actually restrict the size of a, b, and c, it just says that they can't be too far apart from each other. However, by too far apart, we're talking like half the size of each other. This gives a restriction similar to saying that a, b, and c have to give sides that can form a triangle. But there are infinitely many triangles with integer length sides.
(The actual restriction you found is actually even weaker than the triangle thing.)
Say you found a method that only required you to do some sort of test on prime numbers up to the solution. Unfortunately for you, the number of prime numbers under the smallest solution for this problem has almost 80 digits.
I think what I need to convey is that this is a genuinely hard problem. Like the kind of problem that takes years and years of study at the graduate and research level to be able understand and answer. In particular, it's not the kind of thing that like "first instinct" elementary methods are going to work on.
Here is a link to a paper that discusses this and similar problems: https://asu.pure.elsevier.com/en/publications/an-unusual-cubic-representation-problem
Like I saw in a comment you said that you got that c < k(a+b) for some k and similar bounds for a and b.
I made a post where I found a method to iteratively get the bounds really really tight. I don't know how tight exactly they can go as of yet because I didn't bother that much but you can get them really really tight. If c=k(a+b), then at first you restrict k from 2.5 to 4, then the next step gets it to like between 3.2 and 3.7 or something and you keep going. It's possible that it may actually converge to something, I'm not sure, I didn't keep it up. Thing is, that only restricts c in terms of a+b which is a bit of an issue. Now if the simplest solution is like 80 digits long then that's still an issue because these are not good enough restrictions, however I think if I spend a little bit more time I can figure out a few more.
I think what I need to convey is that this is a genuinely hard problem. Like the kind of problem that takes years and years of study at the graduate and research level to be able understand and answer. In particular, it’s not the kind of thing that like “first instinct” elementary methods are going to work on.
It is really hard to solve, yes, but it is generally not as hard to find an assisted brute force method to find SOME solution, even though the general problem may be much much harder.
One issue is that I know next to nothing about computer stuff and I don't know what is feasible and what isn't. But you can certainly restrict your search a bunch.
I recognize there is an issue where the numbers are so damn big that you can restrict like 99.9% of combos you have to check and it may still be too much. But now you're gonna make me go back and try to find more restrictions lol
You're not gonna brute force that problem Pezevenk
I found a lot of other much stronger restrictions that practically completely restrict one of the 3 variables as well as put a bunch of restrictions on the other two and I think it can be done. Or maybe not idk it's still huge.
Like I saw in a comment you said that you got that c < k(a+b) for some k and similar bounds for a and b. This doesn't actually restrict the size of a, b, and c, it just says that they can't be too far apart from each other. However, by too far apart, we're talking like half the size of each other. This gives a restriction similar to saying that a, b, and c have to give sides that can form a triangle. But there are infinitely many triangles with integer length sides.
(The actual restriction you found is actually even weaker than the triangle thing.)
Say you found a method that only required you to do some sort of test on prime numbers up to the solution. Unfortunately for you, the number of prime numbers under the smallest solution for this problem has almost 80 digits.
I think what I need to convey is that this is a genuinely hard problem. Like the kind of problem that takes years and years of study at the graduate and research level to be able understand and answer. In particular, it's not the kind of thing that like "first instinct" elementary methods are going to work on.
Here is a link to a paper that discusses this and similar problems: https://asu.pure.elsevier.com/en/publications/an-unusual-cubic-representation-problem
I made a post where I found a method to iteratively get the bounds really really tight. I don't know how tight exactly they can go as of yet because I didn't bother that much but you can get them really really tight. If c=k(a+b), then at first you restrict k from 2.5 to 4, then the next step gets it to like between 3.2 and 3.7 or something and you keep going. It's possible that it may actually converge to something, I'm not sure, I didn't keep it up. Thing is, that only restricts c in terms of a+b which is a bit of an issue. Now if the simplest solution is like 80 digits long then that's still an issue because these are not good enough restrictions, however I think if I spend a little bit more time I can figure out a few more.
It is really hard to solve, yes, but it is generally not as hard to find an assisted brute force method to find SOME solution, even though the general problem may be much much harder.
One issue is that I know next to nothing about computer stuff and I don't know what is feasible and what isn't. But you can certainly restrict your search a bunch.
I recognize there is an issue where the numbers are so damn big that you can restrict like 99.9% of combos you have to check and it may still be too much. But now you're gonna make me go back and try to find more restrictions lol