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, plus a possible surcharge, 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:
Exq(A,L) = n_L,   or   Exq(A) = {n₁,n₂,...,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 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.