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Fungrim entry: 056c0e

(xQandsin ⁣(πx)Q)    (sin ⁣(πx){0,12,12,1,1})\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
QQQ\mathbb{Q} Rational numbers
Sinsin ⁣(z)\sin\!\left(z\right) Sine
ConstPiπ\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"))

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

2019-08-25 15:30:03.056001 UTC