10.   Glossary

Offset curve   Offset curves are the most general type of curve on this website. Every curve that can be graphed on the spiral is an offset curve. They are called offset curves because they remain a fixed distance from some particular angle. They are generated by quadratic formulas which have a perfect square as the coefficient of x2. If an offset curve is graphed on Cartesian coordinates, it is the non-negative portion of the right limb of a parabola. When graphed on the spiral, as they approach infinity, they become straight and parallel to the angle from which they are offset.

Composite offset curve   Composite offset curves are a subset of offset curves. Their formulas have c equal to zero. Every integer on a composite offset curve is non-prime except for the first, which is zero, and possibly the second, which may or may not be prime. The offset of a composite offset curve is always the square of a fraction.

Product curve   Product curves are a subset of composite offset curves. They are set at a fixed distance from either curve S (the perfect squares) or curve P (the pronics). Speaking more exactly, they are offset from either angle zero or angle 1/2. Their significance is that every factorization involving two numbers is represented on one of these curves. A prime may be defined as a number that is found on only one product curve.  

Curve S   A line drawn through the perfect squares 1, 4, 9, 16, etc. It's generated by the function y = x2. Curve S is the only straight offset curve on the number spiral.  

Curve P   A line drawn through the pronics 2, 6, 12, 20, etc. It's generated by the function y = x2 + x.  

Pronic   A pronic is an integer with factors that differ by one. For example, 1x2 = 2, 2x3 = 6, 3x4 = 12, etc.  

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Copyright © 2003, 2007 Robert Sacks