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Now I and other non-mathematicians can look into Euler's totient function and RSA, and learn more. Thanks for answering.

So is there really never a proof in pure mathematics with more obvious or immediate near-term applications?

Are there any historical examples of that? A major breakthrough in mathematics that once understood, it was immediately obvious that it was going to make X work better, and then it did?

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Mathematics with immediate payoff is usually called physics or computer science. For example the simplex method for linear programming made a lot of things more efficient.

The maths that has an immediate near-term application usually falls, almost by definition, under applied mathematics (or possibly a related field, such as physics, engineering, computer science etc.).

Pure mathematics essentially concerns itself with the mathematics of mathematics. It's basically the process of trying to either prove certain "empirically discovered" mathematical facts, or understand deeply what those facts mean in a more general way. Understanding this type of mathematics in this deep way allows further mathematics to bud off of that specific

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