Show that the fraction \( \frac{21n + 4}{14n + 3} \) is irreducible for all \( n \in \mathbb{N} \).

Prove the divisibility criterion by 17: The number \( \overline{a_n a_{n-1} \ldots a_1 a_0} \) is divisible by 17 if and only if the number \( \overline{a_n a_{n-1} \ldots a_1} - 5a_0 \) is divisible by 17.
Note: This criterion can be used iteratively on smaller numbers. For example: \( 82858 \rightarrow 8285 - 5 \cdot 8 = 8245 \rightarrow 824 - 5 \cdot 5 = 799 \rightarrow 79 - 5 \cdot 9 = 34 \). Since 34 is divisible by 17, the criterion confirms that 82858 is also divisible by 17.

Prove that if \( p \) and \( p^2 + 8 \) are primes, then \( p^3 + 8p + 2 \) is also prime.

How many natural numbers that are multiples of 15 have exactly 15 positive divisors?

Find all pairs of integers \( (x, y) \) satisfying the equation \( x^4 - y^2 = 17 \).

Show that if a prime number \( p \) is divided by 30, then the remainder is either 1 or a prime number. Is the same true when the divisor is 60 or 90?

Let \( m \) and \( n \) be non-zero natural numbers such that \( mn + 1 \) is divisible by 24. Show that \( m + n \) is also divisible by 24.

Let \( a \) and \( b \) be distinct natural numbers. Show that there are infinitely many natural numbers \( s \) such that \( a + s \) and \( b + s \) are coprime.

Give an example of a composite number whose decimal representation contains exactly 1994 digits "1" and one "7" (and no other digits).

Show that there exist no \( x, y \in \mathbb{Q} \) such that \( x^2 + y^2 = 3 \).

Let \( n \in \mathbb{N}_0 \). Show that if \( 2^n - 1 \) is prime, then \( n \) is prime. Show that if \( 2^n + 1 \) is prime, then \( n \) is a power of 2.

For every positive integer \( n \), find the remainder when \( \frac{(7n)!}{7^n \cdot n!} \) is divided by 7.

Find the smallest positive integer \( a \) such that 1971 divides \( 50^n + a \cdot 23^n \) for every odd \( n \).

Determine the greatest common divisor of all numbers of the form \( p^8 - 1 \), where \( p \) is a prime greater than 5.

Find the number of pairs of positive integers \( x, y \) with \( x \leq y \) such that \( \gcd(x, y) = 5! \) and \( \mathrm{lcm}(x, y) = 50! \).

Prove Wilson's theorem: A number \( p \geq 2 \) is prime if and only if \( (p - 1)! \equiv -1 \mod p \).

Find two positive integers \( a \) and \( b \) such that \( (a + b)^7 - a^7 - b^7 \) is divisible by 77 but \( ab(a + b) \) is not divisible by 7.

Find all right triangles where the hypotenuse has an integer length and the other two sides have lengths that are perfect squares.

If \( p \) and \( q \) are prime numbers, and \( x \in \mathbb{Z} \) such that \( q \mid (x^{p-1} + x^{p-2} + \ldots + x + 1) \), then show that \( p = q \) or \( p \) divides \( q - 1 \).

Determine all positive integers \( n \) such that \( 2^n - 1 \cdot n + 1 \) is a perfect square.

Show that there are infinitely many non-zero natural numbers \( n \) such that the number \( 1^2 + 2^2 + \ldots + n^2 / n \) is a perfect square. Clearly, \( 1 \) is the smallest number with this property. Find the next two smallest numbers with this property.

Call an integer \( n \) faithful if there exist positive integers \( a < b < c \) such that \( a \) divides \( b \), \( b \) divides \( c \), and \( n = a + b + c \). Find all unfaithful numbers.

Let \( p_1, p_2, \ldots, p_n \) be distinct prime numbers greater than 3. Show that \( 2p_1p_2\ldots p_n + 1 \) has at least \( 4n \) divisors.