I actually wrote and described it back in 2002 - https://forums.overclockers.com.au/threads/show-off.71630/#p...
I didn't think much of it back when i did it and i'm not a mathematician, just a hobbyist. I guess i should publish my prime hobbies more...
Some accessible examples would be Fermat's Last Theorem or the four color problem.
I'm assuming the researchers do not have the intent of confusing a crystal lattice structure with an actual mathematical lattice, because while they possibly may share similar influences in their models, one is math, the other is physics.
#keepmathpure
There is. Even though most physics research is extremely mathematical and abstract these days, it's still ostensibly grounded in empirical science. Math is not science, it just provides useful tools and insights for studying science. Unlike physics, the disciplinary imperative of math is not to provide us with truths about this world or any other world. Its imperative is to tell us what must follow as a consequence from a given set of assumptions and definitions. This is a very important philosophical dichotomy because it means that even the most lackadaisical, abstract problems in physics (such as moonshine in high energy physics) are still grounded in something "real." Math need not be grounded to anything real; it can be decoupled from what is real or even possible entirely.
> Keeping math "pure" really means keeping it "purely abstract", so it resists any kind of practical application.
I'm not one to be elitist with regards to pure versus abstract mathematics so I sympathize with your point here. That being said, purely abstract mathematics can be extremely useful even if it doesn't ultimately relate to the real world. Consider what G.H. Hardy wrote nearly a century ago in A Mathematician's Apology:
"...both Gauss and lesser mathematicians may be justified in rejoicing that there is [number theory] at any rate...whose very remoteness from ordinary human activities should keep it gentle and clean."
If only Hardy had lived long enough to see his pure and beautiful number theory sullied with the applications to error-correcting codes and cryptography.
Math is how physics is given practical application, if anything, that means science is more abstract than math is.
Physics should not influence math. Math has to be a consequence of it's own axioms, or have it's properties tested against it's own constructions.
Otherwise math falls apart. Then it's useless for physics.
Practical application is wonderful for math. But the direction math is crafted and interpreted matters a lot.
Could you explain what you mean by that?
The difference is physics has reality to test against - observations that can be measured. Math does not have this. Math's only metric against itself is itself.
Obvious first example all primes above 2 are of the form 2x+1. An obvious repeating pattern of primes.
You can take this a small step further. All primes above 6 are of the form 6x+1 or 6x+5. Anything else is a multiple of 2 or 3. Above 6 only 1/3 of numbers are worthy of being considered prime. This is a slightly less obvious example.
A small step further - all primes above 30 are of the form 30x+1, 30x+7, 30x+11, 30x+13, 30x+17, 30x+19, 30x+23 or 30x+29. Anything else is a multiple of 2,3 or 5. So above 30 only 8/30 numbers are worthy of being considered prime. See how we've created a new pattern for the multiple of 2x3x5 to rule out a swath of prime candidates..
I could repeat this each prime found. eg. I could take the common multiple of 2,3,5,7 (210) and create a similar pattern for all numbers above 210 that rules out the repeated multiples of 2,3,5 and 7. (leaving us just 58/210 numbers worthy of being considered prime).
This is why you see peaks of primes at various repeating multiples. For every new prime found you can take the multiple of it and all previous primes. From that you can rule out primality for various offsets to any multiples of that number. So primes above certain numbers can only possibly exist in certain forms. Which is why you see primes at repeated patterns from each other - the primes can only exist in those forms.