One of the best aspects of the sciences is endless challenge, when ideas face relentless scrutiny and if found wanting, are replaced by new, better and more effective ones. But theoretically mathematics is on a different system, and is, with mathematical proof.
Because a mathematical proof is a perfect entity, it cannot be wrong, so it will never collapse or fail.
However, confidence in mathematical proof while logical, does not mean confidence in a mathematical approach is the same thing, and today the world relies on some conjectures for public key encryption, a popular security system with mathematical underpinnings.
The problem is that any public key can be broken if you can factor a certain rather large, by our standards, number, where the understanding for some time has been it is hard to factor such numbers.
But xy = P, where P is a product, may look hard, but is it? There is no mathematical proof that it is, and the belief that if P is sufficiently large then it is difficult to find x and y is at this point in time just a matter of opinion and faith in the abilities of people who have been known to have worked to solve such problems.
Historically such faith has been found to be wanting. It has NOTHING to do with mathematics. It is JUST about confidence in human ability.
I will note that this area is covered by binary quadratic Diophantine equations, as xy = P is one. So yes, I have lots of research and for awhile focused on integer factorization but stopped when I became concerned that maybe I could find much easier ways.
I worried, if I did, could the world handle the consequences? That faith in people shattered in a moment could have unfortunate results. So I moved on to other areas and NOT saying I could find an easier way, as then just emphasizing more faith in a person.
Weirdly enough in this area the notion that the problem hasn't been solved more easily is based on the belief that if anyone had done it, the world would be told.
But if you could easily break into computer systems around the globe using such systems and peruse endless web traffic encrypted with them, with people endlessly believing you couldn't, and even maybe got away with astonishing things as their confidence refused to be shaken, would you tell anybody?
During World War II, the British broke encryption systems, along with my country the US, and did they tell? Yeah, after they'd won the war.