Let , and

Here is a definition of Mahler measure.

Lehmer conjectured,

**Lehmer’s Mahler-measure Conjecture:**

There exists a real number such that or . Lehmer suggested

where is given as

.

Now, let us “try” to construct a polynomial , such that .

First, consider . Hence, ; however, , so we need to calibrate the polynomial with more terms.

Let

where and .

Thus, .

Let us substitute by and by ; then, we may rewrite the polynomial as

Here are the first few cases;

, then

, then

, then

Obviously, for given , and share the same coefficient.

Since the polynomial is getting messier, we need a new notation; let denote the sum of product of -combination from set of . For instance, .

Now, in stead of writing down the whole family of polynomials into a table. Let us define a matrix where each entry , (the -th row and -th column) represents the coefficient of from . The entry is left blank if the coefficient is 0, just to avoid confusion.

The coefficients seem pretty random though, there is a specific pattern lurking.

Consider . Then, term in is only determined by and terms from .

For example, consider ; it is . According to the idea presented above, this term is created by

.

Since the degree of is all 4, let us not consider about the for now. Then,

and

.

Thus, it nicely transforms to the term, as desired.

With this nice property, I induced a following equation.

**Lee Equation**:

As we move back to the matrix, let us assume that has Mahler measure less than . Then, every term along the -th row is composed of integers; obviously it leads to the conclusion that for all are integers.

If Lehmer’s conjecture turns out to be true, then it implies for every , there exists no sets of that renders for every .