Contents

## Exercise 1

Let $$x_{n}$$ be a sequence of positive integers so defined:

$\begin{array}{l} x_{1}= 2 \\ x_{n + 1} = x_{n}^{2} – x_{n} +1 \quad n \gt 1 \\ \end{array}$

Prove that the numbers $$x_{n}$$ are pairwise relatively prime.

Solution
The values of the sequence are obtained by iterating the polynomial $$P (x) = x^{2} -x + 1$$. Reasoning by induction, it can be easily seen that given $$m \lt n$$ the following relation hold true:

$x_{n}= x_{m}Q(x_{m}) +1$

where $$Q (x)$$ is a polynomial with integer coefficients. So $$(x_{n}, x_{m}) = 1$$, as a factor common to $$x_{n}$$ and $$x_{m}$$ should divide the number $$1$$ .

## Exercise 2

Let p be an odd prime number. Prove that the numerator of the following rational number:

$1 + \frac{1}{2} + \frac{1}{3}+ \cdots + \frac{1}{p-1}$

is divisible by $$p$$.

## Exercise 3

Let p be an odd prime number greater than $$3$$. Prove that the numerator of the following rational number:

$1 + \frac{1}{2} + \frac{1}{3}+ \cdots + \frac{1}{p-1}$

is divisible by $$p^{2}$$.

## Exercise 4

Let $$\{p_{1}, p_{2}, \cdots p_{n}, \cdots \}$$ be the ordered sequence of primes. Prove that

$p_{n+1} \le p_{1}p_{2} \cdots p_{n} +1$

## Exercise 5

Prove that for every positive integer $$N$$ there exists a prime number whose sum of decimal places is greater than $$N$$.

For this exercise we use the following important theorem of Dirichlet (1805-1859):

Theorem (Dirichlet)
Each arithmetic progression $$\{an + b, n = 1,2, … \}$$ with $$(a, b) = 1$$ contains infinite primes.
For a proof see for example [1].

Hint for exercise 5
For each $$N \gt 0$$ we have $$(10^{N}, 10^{N} -1) = 1$$. Dirichlet’s theorem assures the existence of a prime number $$p = 10^{N} n + 10^{N} -1$$. Note that the $$N$$ digits of the number $$10​​^{N} -1$$ are all equal to $$9$$.

## Exercise 6

Prove that the last four digits of the numbers $$\{5^{n},\ n = 1,2,3 … \}$$ form a periodic sequence. Find the period.

Solution
The first values ​​of the sequence are the following:

$\{5,25,125,625,3125,15625,78125,390625,1953125\}$

We note that $$5^{n + 4} -5^{n} \equiv 0 \pmod {10000}$$ if $$n \ge 4$$, since $$5^{4} \cdot (5^{4} -1) = 390000$$; therefore the last $$4$$ digits form a period of length $$4$$. The period consists of the numbers $$\{0625,3125,5625,8125 \}$$.

## Exercise 7

Find all the integers $$(m, n)$$ such that $$m^{4} + 4n ^{4}$$ is a prime number.

Hint
Use the following relationship:

$m^{4} + 4n^{4}=[(m+n)^{2}+n^{2}][(m-n)^{2}+n^{2}]$

Then, deduce that it must be $$(mn)^{2} + n^{2} = 1$$ and conclude.

$[m = n = 1]$

## Exercise 8

The Euler function $$f (x) = x^{2} + x + 41$$ takes all prime values ​​for $$x = 0,1,2, \cdots 39$$, as can be verified.
Prove, without making calculations, that it assumes prime values ​​even for negative numbers $$x = -1, -2, \cdots -40$$.

## Exercise 9

Prove the following relationship:

$0 \le \Bigl \lfloor x \Bigr \rfloor – 2 \left \lfloor \frac{x}{2} \right \rfloor \le 1$

In other words, the expression $$\left \lfloor x \right \rfloor – 2 \left \lfloor \frac {x} {2} \right \rfloor$$ takes only the values ​​$$0$$ and $$1$$.

## Exercise 10

Let $$\{p_{1}, p_{2}, \cdots p_{n}, \cdots \}$$ be the ordered sequence of primes. Prove the following inequality:

$p_{n} \lt 2^{2^{n}}$

Hint
Use exercise $$4$$ and proceed by induction.

## Exercise 11

Let us consider $$9$$ distinct positive integers whose prime factors lie in the set $$\{3,7,11\}$$.
Prove that there must be two whose product is a perfect square.

## Bibliography

[1]T. Apostol – Introduction to Analytic Number Theory (Springer-Verlag)