Pi−e

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It is not known whether pi+e or pi-e are rational or not, moreover it is not even known whether pi+e or pi-e are algebraic or not! In fact, there is no pair of non-zero integers $\scriptstyle m\,$ and $\scriptstyle n\,$ for which it is known whether $\scriptstyle m\pi +me\,$ is rational or not. Moreover, it is not known whether the set $\scriptstyle \{\pi ,e\}\,$ is algebraically independent over $\scriptstyle \mathbb {Q} \,$.

Decimal expansion of pi-e

The decimal expansion of pi-e is

$\pi -e=0.42331082513074800310235591192\ldots \,$

giving the sequence of decimal digits (A073244)

{4, 2, 3, 3, 1, 0, 8, 2, 5, 1, 3, 0, 7, 4, 8, 0, 0, 3, 1, 0, 2, 3, 5, 5, 9, 1, 1, 9, 2, 6, 8, 4, 0, 3, 8, 6, 4, 3, 9, 9, 2, 2, 3, 0, 5, 6, 7, 5, 1, 4, 6, 2, 4, 6, 0, 0, 7, 9, 7, 6, 9, 6, 4, 5, 8, 3, 7, 3, 9, 7, 7, 5, 9, 3, 2, 6, 6, 1, 4, ...}

Continued fraction expansion of pi-e

The simple continued fraction expansion of pi-e is

$\pi -e=0~+~{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{18+{\cfrac {1}{7+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}\,$

giving the sequence of partial quotients (A110564)

{0, 2, 2, 1, 3, 6, 18, 7, 1, 10, 1, 3, 1, 39, 6, 1, 1, 7, 1, 2, 1, 7, 2, 1, 1, 1, 5, 29, 14, 7, 2, 1, 1, 4, 1, 11, 932, 1, 2, 1, 1, 2, 10, 3, 1, 4, 1, 1, 4, 1, 2, 7, 1, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 5, 1, 38, 1, 5, 1, 3, 1, 3, 10, ...}

See also

• pi+e
• pi-e
• pi*e
• pi/e
• pi^e
• e^pi (Gelfond's constant)