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De Morgan's theorem is used to help simplify Boolean Expressions. Digital Circuits can be simplified by the application of this theorem.
The laws that let you remove or introduce parentheses in logic expressions."not (a and b)" is the same as "not a or not b" and: "not (a or b)" is the same as "not a and not b" Similar in set theory, with union versus intersection. For more details, check the Wikipedia article "De Morgan's law".
de Moirve's theorem, Pascal's triangle, Pythagoras triangle, Riemann hypothesis, Fermat's last theorem. and many more
Pierre De Fermat is famous for Fermat's Last Theorem, which states that an+bn=cn will never be true as long as n>2
The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic.