## Abstract.

A *length k Newman polynomial* is any polynomial
of the form (where ). Some Newman polynomials are reducible over the rationals,
and some are not. Some Newman polynomials have roots on the unit circle,
and some do not.
Defining, in a natural way, what we mean by the “proportion”
of length

*k*Newman polynomials with a given property, we prove that

of length 3 Newman polynomials are reducible over the rationals,

of length 3 Newman polynomials have roots on the unit circle,

of length 4 Newman polynomials are reducible over the rationals,

of length 4 Newman polynomials have roots on the unit circle.

We also show that certain plausible conjectures imply that the proportion of length 5 Newman polynomials with roots on the unit circle is .

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