91Ó°ÊÓ

Multiply 8023 by 4638 using the method of al-Uqlidis?

Solve \(\frac{1}{2} x^{2}+5 x=28\) by multiplying first by 2 and then using al- Khw?rizmi's procedure. Similarly, solve \(2 x^{2}+\) \(10 x=48\) by first dividing by 2

Solve the following problems of Ab? K?mil: a. Suppose 10 is divided into two parts and the product of one part by itself equals the product of the other part by the square root of 10 . Find the parts. b. Suppose 10 is divided into two parts, each one of which is divided by the other, and the sum of the quotients equals the square root of 5 . Find the parts. (Ab? K?mil solves this in two ways, once directly for \(x\), and a second time by first setting \(y=\frac{10-x}{x}\).)

Complete al-Samaw'al's procedure of dividing \(20 x^{2}+30 x\) by \(6 x^{2}+12\) to get the result stated in the text. Prove that the coefficients of the quotient satisfy the rule \(a_{n+2}=-2 a_{n}\) where \(a_{n}\) is the coefficient of \(\frac{1}{n}\)

Give a complete inductive proof of the result $$ \sum_{i=1}^{n} i^{3}=\left(\sum_{i=1}^{n} i\right)^{2} $$ and compare with al-Karaji's proof.

Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the integers: $$ 1^{5}+2^{5}+\cdots+n^{5}=\frac{1}{6} n^{6}+\frac{1}{2} n^{5}+\frac{5}{12} n^{4}-\frac{1}{12} n^{2} $$

Show, using the formulas for sums of fourth powers and squares, that $$ \begin{aligned} \sum_{i=1}^{n-1}\left(n^{4}-2 n^{2} i^{2}+i^{4}\right) &=\frac{8}{15}(n-1) n^{4}+\frac{1}{30} n^{4}-\frac{1}{30} n \\ &=\frac{8}{15} n \cdot n^{4}-\frac{1}{2} n^{4}-\frac{1}{30} n \end{aligned} $$

Show that one can solve \(x^{3}+d=c x\) by intersecting the hyperbola \(y^{2}-x^{2}+\frac{d}{c} x=0\) with the parabola \(x^{2}=\sqrt{c} y\). Sketch the two conics. Find sets of values for \(c\) and \(d\) for which these conics do not intersect, intersect once, and intersect twice.

Show that \(x^{3}+c x=b x^{2}+d\) is the only one of al-Khayy?mi's cubics that could have three positive solutions. Under what conditions do these three positive solutions exist? How many positive solutions does the equation \(x^{3}+\) \(200 x=20 x^{2}+2000\) have? (The solution of this equation enabled al-Khayy?mi to solve his quadrant problem.)

Show that 1184 and 1210 are amicable numbers that are not a consequence of the theorem of Th?bit ibn Qurra.