Series, and Floors, and Numbers Oh My!!!!

$1^{\circ}$ Finding $P$ .

In order to find $P$ we are first obliged to find $S$ .

Considering $S=\sum_{n=1}^{100} \frac{1}{n}$ , we can easily recognize it to be a harmonic series sum. Although it has been proven that the series diverges, it does that very slowly, which is due to the fact that the partial sums of the series have logarithmic growth.

However, Euler made an approximation of this series, which states that

$\sum_{n=1}^{k} \frac{1}{n} = \ln{k} + \gamma + \epsilon _{k}$ , where $\gamma\approx 0.57721$ is the Euler–Mascheroni constant and $\epsilon_{k}=\frac{1}{2k}$ , which obviously approaches $0$ as $k\rightarrow \infty$ .

Consequently, we are able to approximate the sum of the first $100$ terms.

Therefore, we find

$S=\sum_{n=1}^{100} \frac{1}{n}=\ln{100} + \gamma + \epsilon _{100}\approx \boxed{5.18738}$ (this is the part where we use a calculator)

Now we can easily find $P$ , considering $1000S=5187.38$ and $\lfloor 1000S\rfloor=5187$ .

Thus, we get $\boxed{P=5187}$

$2^{\circ}$ Finding $Q$ .

Let us first look at the equation $a^{3}+b^{3}=c^{3}+d^{3}$ . This is a Diophantine equation which has infinitely many solutions. However, if we are trying to find the smallest positive solution for $Q$ , we should look at $Q$ as a Taxicab number *, especially considering in both equations (namely $a^{3}+b^{3}=Q$ and $c^{3}+d^{3}=Q$ ) Q equals the sum of two cubes. Since we are looking at $2$ equations, it is obvious that we need $Ta(2)$ (more commonly known as the Hardy-Ramanujan number), which equals $1729$ .

Thus, we get $\boxed{Q=1729}$

From $1^{\circ}$ and $2^{\circ}$ we can finally find $Z$ , which is

$Z=\frac{P}{Q}=\frac{5287}{1729}=3$ .

Therefore, the solution is $\boxed{Z=3}$ .

*In mathematics, the $n$ -th Taxicab number (denoted $Ta(n)$ ) is defined as the smallest positive number that can be expressed as the sum of two positive algebraic cubes in $n$ distinct ways. The second Taxicab number in particular is more commonly known as the Hardy-Ramanujan number and is also the reason why these numbers got the name "Taxicab".

The anecdote says that while Ramanujan was lying in hospital sick of tuberculosis, Hardy used to visit him every day. One day, Hardy was taken to the hospital with a taxi numbered $1729$ . He then remarked to his friend that it might be a bad omen since he found the number $1729$ to be rather dull. However, to his amusement, Ramanujan noted that $1729$ was in no case a dull number, but the smallest possible number that can be represented as a sum of two cubes in two distinct ways!

Thus, in honor of Hardy's and Ramanujan's "accidental" discovery, these kinds of numbers are called Taxicab Numbers .

• 1729=Q=1^3+12^3=9^3+10^3 ( i figured this out last time on s.o's post in here)
• With my calculator (CASIO fx-570ES PLUS in my country only, i think) S= 5.187 then P=5187 then Z =P/Q=3