Specific, usually well-known application of a mathematical rule or law

In logic, especially as applied in mathematics, concept A is a **special case** or **specialization** of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A.^{[1]} A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. If B is true, one can immediately deduce that A is true as well, and if B is false, A can also be immediately deduced to be false. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.

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Examples

Special case examples include the following:

- All squares are rectangles (but not all rectangles are squares); therefore the square is a special case of the rectangle.
- Fermat's Last Theorem, that a
^{n} + b^{n} = c^{n} has no solutions in positive integers with n > 2, is a special case of Beal's conjecture, that a^{x} + b^{y} = c^{z} has no primitive solutions in positive integers with x, y, and z all greater than 2, specifically, the case of x = y = z.
- The unproven Riemann hypothesis is a special case of the generalized Riemann hypothesis, in the case that
*χ*(*n*) = 1 for all *n.*
- Fermat's little theorem, which states "if p is a prime number, then for any integer
*a*, then $a^{p}\equiv a{\pmod {p))$" is a special case of Euler's theorem, which states "if *n* and *a* are coprime positive integers, and $\phi (n)$ is Euler's totient function, then $a^{\varphi (n)}\equiv 1{\pmod {n))$", in the case that n is a prime number.
- Euler's identity $e^{i\pi }=-1$ is a special case of Euler's formula which states "for any real number
*x*: $e^{ix}=\cos x+i\sin x$", in the case that x = $\pi$.