Fun numbers
The smallest positive integer that's representable as the sum of two cubes in two different ways
1729
Killed in a duel at 19.
Évariste Galois
The order of a subgroup divides the order of a group
Lagrange's theorem
A property satisfied nowhere by the Weierstrass function despite being uniformly continuous.
Differentiability
The unique solution to the initial value problem f'=f and f(0)=1
The exponential function
The asymptotic difference between the harmonic series and log
The Euler-Mascheroni constant γ
Suicide by cyanide, following legally mandated chemical castration.
Alan Turing
A set has strictly smaller cardinality than its powerset
Cantor's theorem
The topologist's circle (or Warsaw circle) has this property, but not its "local" version.
(path)-connectedness.
A smooth, log-convex (over ℝ+) extension of the factorial to most complex numbers.
Gamma function
The first number proven to be transcendental
Liouville's constant
Killed by a soldier on the beach while doing geometry
Archimedes
Any continuous map from a closed disc to itself has a fixed point.
Brouwer's fixed-point theorem.
A number described by a prominent mathematician as prime, despite being expressible as a product of integers greater than 1.
57
A nonelementary function providing a very good estimate of the prime counting function.
Logarithmic integral Li
Apocryphally, the number Paul Erdos suggested that if an alien civilization threatened to destroy earth unless we can compute it, then we should put all of our combined resources into destroying the aliens.
R(6,6)
Refused to eat out of fear of being poisoned, and subsequently died of malnutrition.
Kurt Gödel
The only social choice function that is both Pareto efficient and independent of irrelevant alternatives is a dictatorship.
Arrow's theorem
The Ackerman Function is computable, but not this.
Primitive-recursive.
Determines the maximum number of steps a 2-symbol Turing machine with 69 states can possibly run before halting
Busy Beaver 69
Legendre's constant - related to the asymptotic distribution of prime numbers.
What is one?
Killed in action fighting for Germany in WWII
Oswald Teichmüller
The axiom of choice implies the law of the excluded middle
Diaconescu's theorem
The variety described in Appendix B example 3.4.1 in Hartshorne has this property, despite not being projective.
Completeness
A function mapping the continued fraction expansion of a number to consecutive runs of a digit in its binary expansion
(Minkowski's) ? function