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van Roomen's Equation

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The van Roomen equation is the degree-45 polynomial equation posed as a challenge by Adriaan van Roomen (1593). Write

P(x)=2T_(45)(x/2)
(1)
=x^(45)-45x^(43)+...+95634x^5-3795x^3+45x,
(2)

where T_(45) is a Chebyshev polynomial of the first kind. Under the substitution x=2sintheta, it satisfies the multiple-angle formula P(2sintheta)=2sin(45theta).

Van Roomen's 1593 challenge was to solve

P(x)=b,
(3)

for the specific algebraic number

b=2sin(pi/(15))
(4)
=sqrt(7/4-(sqrt(5))/4-sqrt((15)/8-(3sqrt(5))/8)).
(5)

For this value, the equation P(x)=b has 45 real roots, given by x=2sin(pi/(675)+(2pik)/(45)) for k=0, 1, ..., 44. Of these roots, 23 are positive and 22 are negative. François Viète found all 23 positive solutions; the smallest is x=2sin(pi/(675)) (Viète 1595, van Assche 2022).

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. New York: Springer-Verlag, p. 108, 2010. 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.Viète, F. Ad problema quod omnibus mathematicis totius orbis construendum proposuit Adrianus Romanus. Paris, France: Jamet Mettayer, 1595. Reprinted in Opera Mathematica (Ed. F. van Schooten). Leiden, Netherlands, pp. 305-323, 1646.

Cite this as:

Weisstein, Eric W. "van Roomen's Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/vanRoomensEquation.html

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