Special case

In

is a special case which is in some way qualitatively different from almost all of the cases allowed.

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.
• 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 ${\displaystyle a^{p}\equiv a{\pmod {p}}}$" is a special case of
  coprime
  positive integers, and ${\displaystyle \phi (n)}$ is Euler's totient function, then ${\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}}$", in the case that n is a prime number.
• Euler's identity ${\displaystyle e^{i\pi }=-1}$ is a special case of Euler's formula which states "for any real number x: ${\displaystyle e^{ix}=\cos x+i\sin x}$", in the case that x = ${\displaystyle \pi }$.

References