Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
See all formats and pricing
More options …
Volume 24, Issue 4


Niven’s Theorem

Artur Korniłowicz / Adam Naumowicz
Published Online: 2017-02-23 | DOI: https://doi.org/10.1515/forma-2016-0026


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].

Keywords: Niven’s theorem; rational root theorem; integral root theorem

MSC 2010: 97G60; 12D10; 03B35


  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Google Scholar

  • [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Google Scholar

  • [3] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.Google Scholar

  • [4] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.Google Scholar

  • [5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Google Scholar

  • [6] Yuzhong Ding and Xiquan Liang. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(3):243–246, 2004.Google Scholar

  • [7] Magdalena Jastrzębska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307–313, 2004.Google Scholar

  • [8] J.D. King. Integer roots of polynomials. The Mathematical Gazette, 90(519):455–456, 2006. doi:http://dx.doi.org/10.1017/S0025557200180295Crossref

  • [9] Serge Lang. Algebra. Addison-Wesley, 1980. Google Scholar

  • [10] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391–395, 2001.Google Scholar

  • [11] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461–470, 2001.Google Scholar

  • [12] Ivan Niven. Irrational numbers. The Carus Mathematical Monographs, No. 11. The Mathematical Association of America. Distributed by John Wiley and Sons, Inc., New York, N.Y., 1956. Google Scholar

  • [13] Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1): 49–58, 2004.Google Scholar

  • [14] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Google Scholar

  • [15] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Google Scholar

  • [16] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Google Scholar

About the article

Received: 2016-12-15

Published Online: 2017-02-23

Published in Print: 2016-12-01

Citation Information: Formalized Mathematics, Volume 24, Issue 4, Pages 301–308, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.1515/forma-2016-0026.

Export Citation

© 2016 Artur Korniłowicz et al., published by De Gruyter Open. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License. BY-SA 3.0

Comments (0)

Please log in or register to comment.
Log in