Fungrim entry: 056c0e

\left(x \in \mathbb{Q} \,\mathbin{\operatorname{and}}\, \sin\!\left(\pi x\right) \in \mathbb{Q}\right) \implies \left(\sin\!\left(\pi x\right) \in \left\{0, \frac{1}{2}, -\frac{1}{2}, 1, -1\right\}\right)

References:

Niven's theorem

TeX:

\left(x \in \mathbb{Q} \,\mathbin{\operatorname{and}}\, \sin\!\left(\pi x\right) \in \mathbb{Q}\right) \implies \left(\sin\!\left(\pi x\right) \in \left\{0, \frac{1}{2}, -\frac{1}{2}, 1, -1\right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`QQ`	$\mathbb{Q}$	Rational numbers
`Sin`	$\sin\!\left(z\right)$	Sine
`ConstPi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("056c0e"),
    Formula(Implies(And(Element(x, QQ), Element(Sin(Mul(ConstPi, x)), QQ)), Element(Sin(Mul(ConstPi, x)), Set(0, Div(1, 2), Neg(Div(1, 2)), 1, -1)))),
    References("Niven's theorem"))

Topics using this entry

Sine

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC