# Modular Equations and Approximations to π

• Lennart Berggren
• Jonathan Borwein
• Peter Borwein

## Abstract

If we suppose that
$$(1 + {e^{ - \pi Nn}})(1 + {e^{ - 3\pi Nn}})(1 + {e^{ - b\pi Nn}}) \ldots = {2^{\frac{1}{4}}}{c^{ - \pi Nn/24}}{G_n} \ldots \ldots \ldots$$
(1)
and
$$(1 - {e^{ - \pi Nn}})(1 - {e^{ - 3\pi Nn}})(1 - {e^{ - b\pi Nn}}) \ldots = {2^{\frac{1}{4}}}{e^{ - \pi Nn/24}}{g_n}, \ldots \ldots \ldots$$
(2)
then G n and g n can always be expressed as roots of algebraical equations when n is any rational number. For we know that
$$(1 + q)(1 + {q^3})(1 + {q^5}) \ldots = {2^{\frac{1}{6}}}{q^{\frac{1}{{24}}}}{(kk')^{ - \frac{1}{{12}}}} \ldots \ldots \ldots \ldots \ldots$$
(3)
and
$$(1 - q)(1 - {q^3})(1 - {q^5}) \ldots = {2^{\frac{1}{6}}}{q^{\frac{1}{{24}}}}{k^{ - \frac{1}{{12}}}}{k'^{\frac{1}{6}}}. \ldots \ldots \ldots \ldots \ldots$$
(4)

## Keywords

Rational Number Arithmetical Progression Algebraic Function Modular Equation Finite Term
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.