Solving Van Roomen's Problem: So far

So, up to know, I've shown that Van Roomen's monster equation: (see here):

(45)x - (3,795)x³ + (95,634)x⁵ - (1,138,500)x⁷ + (7,811,375)x⁹ - (34,512,075)x¹¹ + (105,306,075)x¹³ - (232,676,280)x¹⁵ + (384,942,375)x¹⁷ - (488,494,125)x¹⁹ + (483,841,800)x²¹ - (378,658,800)x²³ + (236,030,652)x²⁵ - (117,679,100)x²⁷ + (46,955,700)x²⁹ - (14,945,040)x³¹ + (3,764,565)x³³ - (740,259)x³⁵ + (111,150)x³⁷ - (12,300)x³⁹ + (945)x⁴¹ - (45)x⁴³ + x⁴⁵ = A

simplifies to this one when n = 45 (see here for proof):



If we generalize the above equation to F_n (see below), there is an interesting recurrence relation that emerges (see here for details).

If we define:

F_n(x) =



Then:

F_n(x) = x*F_n-1(x) - F_n-2(x).

Finally, we can introduce trigonometry to the equation defined above (see here for proof) to get:

for any integer n ≥ 1:

2cos(nα) = F_n(2cosα)

For any odd integer n ≥ 1:

2 sin(nα) = (-1)^(n-1)/2*F_n(2sinα)


Now, let's see if we can simplify the examples that Van Roomen gives for x and A (see here for details). It turns out that all can be simplified using trigonometric identities.

Here are three examples that Van Roomen provides. I should note that in Van Roomen's original problem, he made a mistake on example 2. For purposes of this blog, I have corrected his mistake. (See Jean-Pierre Tignol's book for more details).

Example 1:

If



Then:



Example 2:

If:



Then:



Example 3:

If:



Then:




In my next blog, I will show how each of these values can be simplified using trigonometry.

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001