Beraha constants

The Beraha constants are a series of mathematical constants by which the ${\displaystyle n{\text{th}}}$ Beraha constant is given by

${\displaystyle B(n)=2+2\cos \left({\frac {2\pi }{n}}\right).}$

Notable examples of Beraha constants include ${\displaystyle B(5)}$ is ${\displaystyle \varphi +1}$, where ${\displaystyle \varphi }$ is the golden ratio, ${\displaystyle B(7)}$ is the silver constant (also known as the silver root), and ${\displaystyle B(10)=\varphi +2}$.

The following table summarizes the first ten Beraha constants.

${\displaystyle n}$	${\displaystyle B(n)}$	Approximately
1	4
2	0
3	1
4	2
5	${\displaystyle {\frac {1}{2}}(3+{\sqrt {5}})}$	2.618
6	3
7	${\displaystyle 2+2\cos({\tfrac {2}{7}}\pi )}$	3.247
8	${\displaystyle 2+{\sqrt {2}}}$	3.414
9	${\displaystyle 2+2\cos({\tfrac {2}{9}}\pi )}$	3.532
10	${\displaystyle {\frac {1}{2}}(5+{\sqrt {5}})}$	3.618

In October 2026, Reza Mactabi presented a proof in an arXiv preprint that every non-integer Beraha number is an accumulation point of real roots of the chromatic polynomials of simple plane triangulations. Wolfram MathWorld reports this result as proved by Mactabi.

See also

• Chromatic polynomial

Notes

References

• Weisstein, Eric W. "Beraha Constants". Wolfram MathWorld. Retrieved November 3, 2018.
• Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.
• Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983.
• Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 160–163, 1986.
• Tutte, W. T. "Chromials." University of Waterloo, 1971.
• Tutte, W. T. "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case ${\displaystyle \lambda =1}$," Research Report COPR 72–7, University of Waterloo, 1972a.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case ${\displaystyle \lambda =\infty }$." Research Report COPR 72–4, University of Waterloo, 1972b.

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