91Ó°ÊÓ

Twenty persons, men and women, dine at a tavern. The share of the bill for one man is \(\$ 8\), for one woman \(\$ 7\), and the entire bill amounts to \(\$ 145\). Required, the number of men and women separately. (This exercise and the next two are from Euler's Introduction to Algebra.)

Factor Leibniz's polynomial \(x^{4}+a^{4}\) into two real quadratic polynomials. ( Hint: Add and subtract \(2 a^{2} x^{2}\).)

Factor \(x^{5}-1\) into linear and real quadratic factors.

Use Euler's procedure from his proof that all real quartics factor to determine the factorization of \(x^{4}-2 x^{2}+8 x-3\) as a product of two quadratic polynomials.

Determine the three roots \(x_{1}, x_{2}, x_{3}\) of \(x^{3}-6 x-9=0\) Use Lagrange's procedure to find the sixth-degree equation satisfied by \(y\), where \(x=y+2 / y\). Determine all six solutions of this equation and express each explicitly as \(\frac{1}{3}\left(x^{\prime}+\omega x^{\prime \prime}+\omega^{2} x^{\prime \prime \prime}\right)\), where \(\left(x^{\prime}, x^{\prime \prime}, x^{\prime \prime \prime}\right)\) is a permutation of \(\left(x_{1}, x_{2}, x_{3}\right)\) and \(\omega\) is a complex root of \(x^{3}-1=0\).

In Euler's proof of the case \(n=3\) of Fermat's Last Theorem, we now consider the situation where \(p=3 r\). We then know that \(\frac{3}{4} r\left(9 r^{2}+3 q^{2}\right)=\frac{9}{4} r\left(3 r^{2}+q^{2}\right)\) must be a cube. Show that the two factors in this expression are relatively prime. It follows that each must be a cube. In particular, \(q^{2}+3 r^{2}\) must be a cube. Factor this expression as in the text, using complex numbers of the form \(a+b \sqrt{-3}\), and conclude that \(q=t\left(t^{2}-9 u^{2}\right), r=3 u\left(t^{2}-u^{2}\right)\), where \(t\) is odd and \(u\) is even. Also, since \(\frac{9}{4} r\) is a cube, show that \(\frac{2}{3} r=2 u(t+\) u) \((t-u)\) is a cube where the factors are relatively prime. Conclude as in the case detailed in the text that we can now find three integers smaller than the original set for which the sum of their cubes is a cube.

Determine the quadratic residues modulo 13 .

Prove that \(-1\) is a quadratic residue with respect to a prime \(q\) if and only if \(q \equiv 1(\bmod 4)\).

Benjamin Banneker was fond of solving mathematical puzzles and recorded many in his notebook, including his own version of the old hundred fowls problem: A gentleman sent his servant with \(£ 100\) to buy 100 cattle, with orders to give \(£ 5\) for each bullock, 20 shillings for each cow, and 1 shilling for each sheep. (Recall that 20 shillings equals \(£ 1\).) What number of each sort of animal did he bring back to his master? \(?^{23}\)

Divide 60 into four parts such that the first increased by 4, the second decreased by 4, the third multiplied by 4, and the fourth divided by 4 shall each equal the same number (Banneker).