On Rings of Numbers Which Are Normal To One Base But Non-normal to Another Original Research Article

For any natural number g ≥ 2, and for any odd prime p not dividing g, we give an explicit example of a ring R of real numbers with the Following three properties: • is uncountable, • all numbers set membership, variant , ≠ 0, are normal to base , and • all numbers set membership, variant are non-normal to base · . This result contains, e.g., explicitly given numbers x such that, for any non-constant polynomial q set membership, variant image[x], the number q(x) is normal to base 3 but non-normal to base 15.