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Niven’s Theorem

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Korniłowicz A., Naumowicz A.

Formalized Mathematics

2016

tom 24

nr 4

301-308

This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].

Comments on the height reducing property

84%

Akiyama S., Zaimi T.

Open Mathematics

2013

tom 11

nr 9

1616-1627

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.

Ograniczanie wyników

1 Formalized Mathematics

1 Open Mathematics

1 Akiyama S.

1 Korniłowicz A.

1 Naumowicz A.

1 Zaimi T.

1 2016

1 2013

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Niven’s Theorem

Comments on the height reducing property