The History of Mathematics: A Brief Course

QUESTIONS AND PROBLEMS 185

The first few partial fractions here give

1

Ú

1 +

2+T = 3,

1 2 8

2 + —Ã =^23 = 3 =^2 -^666 -

2

1 _ ~ 4

2 + L— = ^=2.75,

1 +

1 5 19

2 + = = 2- = — = 2.714285712485.

1 + 1 7 7

2 +

1

1 23 87

(^2) + i = (^2) +— = — = 2.71875,
1 +
32 32
2 +
1 + ^
so that the approximations get better and better. Do the same with ð ss 3.14159265,
and calculate the first five approximate fractions. Do you recognize any of these?
7.13. Can the pair of amicable numbers 1184 and 1210 be constructed from Thabit
ibn-Qurra's formula?
7.14. Solve the generalized problem stated by Matsunaga of finding an integer Í
that is simultaneously of the form x^2 + a\X + bi and y^2 + 0,2V + 62- To do this, show
that it is always possible to factor the number (a^2 + 4&i) — (a^2 + 4i>2) as a product
mn, where m and ç are either both even or both odd, and that the solution is
found by taking χ = - á÷), y = ±(!ø- - á 2 ).
7.15. Leonardo's solution to the problem of finding a second pair of squares having
a given sum is explained in general terms, then illustrated with a special case. He
considers the case 4^2 + 5^2 = 41. He first finds two numbers (3 and 4) for which the
sum of the squares is a square. He then forms the product of 41 and the sum of
the squares of the latter pair, obtaining 25 · 41 = 1025. Then he finds two squares
whose sum equals this number: 31^2 and 8^2 or 32^2 and l^2. He thus obtains the
results (f)^2 + (|)^2 = 41 and (f )^2 + (|)^2 = 41. Following this method, find
another pair of rational numbers whose sum is 41. Why does the method work?
7.16. If the general term of the Fibonacci sequence is a„, show that an < an+i <
2an, so that the ratio (Àç+é/á„ always lies between 1 and 2. Assuming that this
ratio has a limit, what is that limit?
7.17. Suppose that the pairs of rabbits begin to breed in the first month after they
are born, but die after the second month (having produced two more pairs). What
sequence of numbers results?
7.18. Prove that if ,x, y, and æ are relatively prime integers such that x^2 +y^2 = z^2 ,
with χ and æ odd and y even, there exist integers u and í such that χ = u^2 - õ^2 ,