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## Integermania!

#### Theorems and Conjectures

Theorems and conjectures about Integermania can be
submitted to Steve Wilson for posting on
this page. Warning: False conjectures could be bumped off this page
by true conjectures!

**Definition 1:** The **exquisiteness of a solution** to a
Integermania problem is the highest level operation
used in its construction, as given by the table of
levels of exquisiteness, but the table is not
repeated here.

**Definition 2:** The **exquisiteness of a set** of numbers $A$ at level $L$ is
the largest positive integer $n_L$ having the property that for every
positive integer less than or equal to $n_L$ there exists a solution to the
Integermania problem using set $A$ with exquisiteness level
less than $L+1$. Notationally, we can write:

$\operatorname{Exq}(A,L) = n_L$, or
$\operatorname{Exq}(A) = \{n_1, n_2, ..., n_L, ...\}$.

See Exquiteness of Sets for conjectures about
different Intergermania problems.

**Theorem 1:** In an Integermania problem with
$n$ values and $k$ available binary operations, where each value is used exactly once,
and values are combined by using only the available binary operations, there are at most
$\dfrac{k^{n-1}(2n-2)!}{(n-1)!}$ positive integers which can be created.
See discussion and proof.

**Theorem 2:** In an Integermania problem
where set $A$ has $n$ values, the level 1 exquisiteness of $A$ will be
less than or equal to 1 when $n=1$, and less than or equal to 3
when $n=2$. See proof and comments
about larger values of $n$.

**Theorem 3:** There exists an Integermania solution,
without rounding, for every integer $n$ and every non-empty set of integers $A$.
See proof and comments.