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There's something quite interesting about the problems in number theory especially. The questions/relationships sometimes don't seem useful at all and are later

There's something quite interesting about the problems in number theory especially. The questions/relationships sometimes don't seem useful at all and are later proven to be incredibly useful. Number Theory is the prime example of this. I believe there's a G H Hardy quote somewhere, about Number Theory being obviously useless, but could only find it from one secondary source, although it does track with his views expressed in A Mathematician's Apology[1] - "The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics." You can find relationships between ideas or topics that are seemingly unrelated, for instance, even perfect numbers and Mersenne primes have a 1:1 mapping and therefore they're logically equivalent and a proof that either set is either infinite or finite is sufficient to prove the other's relationship with infinity. There's little to no intuitive relationship between these ideas, but the fact that they're linked is somewhat humbling - a fun quirk in the fabric of the universe, if you will. [1] https://en.wikipedia.org/wiki/A_Mathematician's_Apology https://en.wikipedia.org/wiki/A_Mathematician's_Apology

vlovich123 2y ago
> No one has yet found any war-like purpose to be served by the theory of numbers or relativity or quantum mechanics, and it seems very unlikely that anybody will do so for many years. G.H.Hard. Eureka, issue 3, Jan 1940
- l33t7332273 2y ago
  Less than 100 years later, we stand waiting for nuclear bombs guided by GPS to be launched when the authorization cryptographic certificate is verified.
  - eigenket 2y ago
    Nuclear weapons (based on quantum mechanics and special relativity) were used less than 6 years after that quote.
    - vlovich123 2y ago
      And number theory was critical to breaking enigma. So they were all used within 5 years.
      - l33t7332273 2y ago
        Was it? My understanding was that they didn’t use number theoretic approaches.
        vlovich123 2y ago
        My bad. You’re right. It was group theory and cryptanalysis. Number theory comes in later in the 1970s for public key cryptography (1976 publicly, early 1970s at GCHQ). So the military work on it really started in the late 1960s.
Xcelerate 2y ago
Number theory appears to be at the core of computability, via Matiyasevich's theorem, which links the behavior of every Turing machine to the solutions (or lack thereof) of Diophantine equations. It’s not surprising to me that number theory constantly ends up being more useful than suspected since in many ways the study of these equations is equivalent to the study of computation.

vlovich123 2y ago

> No one has yet found any war-like purpose to be served by the theory of numbers or relativity or quantum mechanics, and it seems very unlikely that anybody will do so for many years. G.H.Hard. Eureka, issue 3, Jan 1940

l33t7332273 2y ago
Less than 100 years later, we stand waiting for nuclear bombs guided by GPS to be launched when the authorization cryptographic certificate is verified.
- eigenket 2y ago
  Nuclear weapons (based on quantum mechanics and special relativity) were used less than 6 years after that quote.
  - vlovich123 2y ago
    And number theory was critical to breaking enigma. So they were all used within 5 years.
    - l33t7332273 2y ago
      Was it? My understanding was that they didn’t use number theoretic approaches.
      - vlovich123 2y ago
        My bad. You’re right. It was group theory and cryptanalysis. Number theory comes in later in the 1970s for public key cryptography (1976 publicly, early 1970s at GCHQ). So the military work on it really started in the late 1960s.

Xcelerate 2y ago

Number theory appears to be at the core of computability, via Matiyasevich's theorem, which links the behavior of every Turing machine to the solutions (or lack thereof) of Diophantine equations. It’s not surprising to me that number theory constantly ends up being more useful than suspected since in many ways the study of these equations is equivalent to the study of computation.