The van Roomen equation is the degree-45 polynomial
equation posed as a challenge by Adriaan van Roomen (1593). Write
where Chebyshev polynomial of the
first kind . Under the substitution multiple-angle
formula
(3)
where algebraic number . For the main
challenge,
(4)
For this value, real roots . Equivalently, the equation positive
and 22 are negative . François Viète found
all 23 positive solutions; the smallest is
See also Chebyshev Polynomial of the First Kind ,
Multiple-Angle Formulas ,
Polynomial Equation
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References Stillwell, J. Mathematics and Its History, 3rd ed. https://doi.org/10.1007/978-1-4419-6053-5 . Van
Assche, W. "Chebyshev Polynomials in the 16th Century." J. Approx. Theory 279 ,
105767, 2022. https://doi.org/10.1016/j.jat.2022.105767 . van
Roomen, A. Ideae Mathematicae Pars Prima, Sive Methodus Polygonorum. Antwerp:
Ioannem Keerbergium, 1593. https://books.google.com/books?id=nMUfMvJNYl4C .
Cite this as:
Weisstein, Eric W. "van Roomen's Equation."
From MathWorld https://mathworld.wolfram.com/vanRoomensEquation.html
Subject classifications
is a Chebyshev polynomial of the
first kind. Under the substitution 
, it satisfies the multiple-angle
formula 
.
The equation posed by van Roomen is
is a specified algebraic number. For the main
challenge,
,
and the polynomial 
has 45 real roots. Equivalently, the equation 
has 45 real solutions, given by 
for 
, 1, ..., 44. Of these solutions, 23 are positive
and 22 are negative. François Viète found
all 23 positive solutions; the smallest is 
(Van Assche 2022).
