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Due to Euler's prolific output, there are a great number of theorems that are know by the name "Euler's theorem." A sampling of these are Euler's displacement theorem for ...

Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of ...

An elliptic curve of the form y^2=x^3+n for n an integer. This equation has a finite number of solutions in integers for all nonzero n. If (x,y) is a solution, it therefore ...

There are a great many beautiful identities involving q-series, some of which follow directly by taking the q-analog of standard combinatorial identities, e.g., the ...

Define the Euler measure of a polyhedral set as the Euler integral of its indicator function. It is easy to show by induction that the Euler measure of a closed bounded ...

An odd alternating permutation number, more commonly called an Euler number or secant number.

An identity in calculus of variations discovered in 1868 by Beltrami. The Euler-Lagrange differential equation is (partialf)/(partialy)-d/(dx)((partialf)/(partialy_x))=0. (1) ...

Binet's formula is an equation which gives the nth Fibonacci number as a difference of positive and negative nth powers of the golden ratio phi. It can be written as F_n = ...

A reflection relation is a functional equation relating f(-x) to f(x), or more generally, f(a-x) to f(x). Perhaps the best known example of a reflection formula is the gamma ...

Let a prime number generated by Euler's prime-generating polynomial n^2+n+41 be known as an Euler prime. (Note that such primes are distinct from prime Euler numbers, which ...