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Hilbert once said, “The art of doing mathematics consists in finding that special case which contains all the germs of generality.”

What would be (relatively) simple examples?

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    $\begingroup$ The Fourier transform on $\mathbb{R}$ probably falls into this category. Retooling some of the basic ideas makes it pretty easy to generalize to any LCA group. $\endgroup$ – Cameron Williams May 30 '16 at 23:05
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$$x_{n-1} =\frac{\alpha+\beta x_n +\gamma x_{n-1} +\delta x_{n-2}}{A+Bx_{n}+Cx_{n-1}+Dx_{n-2}}, n= 0, 1, ...,$$

where the parameters $\alpha, \beta, \gamma, \delta, A, B,C, D$ are non-negative real numbers and the initial conditions $m$ are arbitrary non-negative real numbers such that the denominator is always positive.

We are primarily concerned with the boundedness nature of solutions, the stability of the equilibrium points, the periodic character of the equation, and with convergence to periodic solutions including periodic trichotomies.

If we allow one or more of the parameters in the equation to be $0$, then we can see that the equation contains

$$(2^4 -1)(2^4 -1)= 225 $$

special cases, each with positive parameters and positive or non-negative initial conditions.

According to David Hilbert “The art of doing mathematics consists in finding that special case which contains all the germs of generality" and according to Paul Halmos 'The source of all good mathematics is the special case, the concrete example"

The special case of this equation contains a lot of the germs of generality of the theory of difference equations of order greater than one about which, at the beginning of the third millennium, we know no surprisingly little. The mathematics behind the special cases of this equation is also beautiful, surprising, and interesting.

The methods and techniques we develop to understand the dynamics of various special cases of rational difference equations and the theory that we obtain will also be useful in analysing the equation in any mathematical model that involves difference equation.

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  • $\begingroup$ @MatemáticosChibchas Oops, it was meant to say "non-negative". I have edited to fix! Thanks for the pickup. $\endgroup$ – anonymous May 31 '16 at 8:17
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Integers. Many theorems of rings, for example, give meaningful non-trivial results. Think of the abstract theory of prime ideals. Take matrices as a generalization - and even that can be considered a nice "special case" in itself. Then think operator algebras... I mean so many structures in algebra just generalize what you may do with individual numbers.

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Pierre de Fermat observed that (in modern notation) $\int_a^bx^n\; dx=(b^{n+1}-a^{n+1})/(n+1)$ for non-negative integer $n$ but not the generality (the Fundamental Theorem of Calculus), found later, independently by Newton and Leibnitz.

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Take a positive definite quadratic form with integer matrix. For example, $$w^2+x^2+y^2+z^2$$ or $$w^2+2x^2+5y^2+5z^2$$ It turns out that if this form represents each positive integer from 1 to 15, then it represents all positive integers. This 15 theorem was first proved by Conway and Schneeberger in 1993, then again (much more simply and beautifully) by Manjul Bhargava in 2000.

Note: For the first example above, the answer is "yes"; for the second example, the answer is "no" -- it takes on every positive integer except 15.

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