What do mathematicians mean when they say some conjecture can’t be proven using the current technology?

This doesn't have any formal meaning. It just means that they believe the problem can't be solved with the techniques that mathematicians have already developed and instead some big new idea will be necessary to solve it. That is, "the current technology" refers to the collection of proof methods that mathematicians have discovered already. It's meant as a sort of metaphor between mathematics and engineering: some engineering problems can be solved by just finding a clever way to put together already existing technologies, while others require major new inventions. In the same way, some mathematics problems can be solved by just finding clever new uses of ideas that are already known while other mathematics problems require something more novel.

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