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If the first case of Fermat's last theorem is false for the prime exponent p, then 3^(p-1)=1 (mod p^2).

A specific type of ultraproduct that can be used to construct nonstandard universes and obtain the transfer principle as a corollary of Łoś' theorem for ultraproducts.

Vizing's theorem states that a graph can be edge-colored in either Delta or Delta+1 colors, where Delta is the maximum vertex degree of the graph. A graph with edge chromatic ...

The 10_3 configuration of ten lines intersecting three at a time in 10 points which arises in Desargues' theorem. Its Levi graph is the Desargues graph.

The Diophantine equation x^n+y^n=z^n. The assertion that this equation has no nontrivial solutions for n>2 has a long and fascinating history and is known as Fermat's last ...

A conjecture which relates the minimal elliptic discriminant of an elliptic curve to the j-conductor. If true, it would imply Fermat's last theorem for sufficiently large ...

The fundamental theorem(s) of calculus relate derivatives and integrals with one another. These relationships are both important theoretical achievements and pactical tools ...

If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Frequently, mathematicians seek to prove the existence of ...

A q-analog of the Saalschütz theorem due to Jackson is given by where _3phi_2 is the q-hypergeometric function (Koepf 1998, p. 40; Schilling and Warnaar 1999).

In the plane, if a line intersects one side of a triangle and misses the three vertices, then it must intersect one of the other two sides. This is a special case of the ...