The reverse-and-add process produces the sum of a number and the number formed by reversing the order of its digits. For example, 56 + 65 = 121. As another example, 125 + 521 = 646.Some numbers become ...
The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because ...
In mathematics, a quasiperfect number is a theoretical natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Quasiperfect numbers are abundant numbers ...
The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + ...
It is unknown whether there are any odd perfect numbers, though various results have been obtained. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. ...
The search for expansions of rational numbers as sums of unit fractions dates to the mathematics of ancient Egypt, in which Egyptian fraction expansions of this type were used as a notation for record ...
The abc conjecture can be expressed as follows: For every ε 0, there are only finitely many triples of coprime positive integers a + b = c such that c d1+ε, where d denotes the product of the dist ...
Dawn of arithmetic The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 (Larsa, Mesopotamia, ca. 1800 BCE) contains a list of "Pythagorean ...
O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set M in a set-theoretic manner, as a sequence S = (Sn), n = 0,1,2,... such thatSn is a b ...
Oswald Veblen in 1904 defined a theory to be categorical if all of its models are isomorphic. It follows from the definition above and the L?wenheim–Skolem theorem that any first-order theory with a ...
The words "process" and "finite number of operations" have been taken to mean that Hilbert was asking for an algorithm. The term "rational integer" simply refers to the integers, positive, negative or ...
As with the concept of a decidable set, the definition of a decidable theory or logical system can be given either in terms of effective methods or in terms of computable functions. These are generall ...
\langle K, \prec_K\rangle, for K a class of structures in some language L, is an AEC if it has the following properties:\prec_K is a partial order on K.If M\prec_K N then M is a substructure of N.Isom ...
For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their valences, so that the object is naturally a σ-struct ...
T will be a complete theory in some language.T is called κ-stable (for an infinite cardinal κ) if for every set A of cardinality κ the set of complete types over A has cardinality κ.ω-stable is a ...
