Prime numbers are important for cryptography - if there is a pattern to them it might have an effect on what we think is secure or not.

https://www.abc.net.au/news/science/2018-01-20/how-prime-num...

Elusiveness is subjective, but there's certainly plenty of patterns to be found (though, to be fair, what constitutes a pattern might also be subjective). You should check out Terence Tao's blog [0], he's not only one of the most prominent mathematicians in the field but he's also excellent at explaining things. These slides [1] for example don't require any background in math.

[0] https://terrytao.wordpress.com

[1] https://www.math.ucla.edu/~tao/preprints/Slides/primes.pdf

The answer to these questions is the Riemann hypothesis. John Baez just wrote a related paper about "motives" for beginners (undergrads?) [1]. It's worth a read even if you squint at the equations like i do, there's plenty of interesting commentary as well.

[1] https://math.ucr.edu/home/baez/motives.pdf

Maybe.

Yes

No.