Prove that the Catalan numbers satisfy the recurrence relationship \(C_n = C_0C_{n-1} + C_1C_{n-2} + \dots + C_{n-1}C_0\). The definition of the Catalan numbers \(C_n\) is given in the handbook.
Determine all prime numbers \(p\) and \(q\) such that \(p^2 - 2q^2 = 1\) holds.
Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of \(d\). Prove that \(d >30,000\).
Could it be that a) \(\sigma(n) > 3n\); b) \(\sigma(n) > 100n\)?
Prove that for a real positive \(\alpha\) and a positive integer \(d\), \(\lfloor \alpha / d\rfloor = \lfloor \lfloor \alpha\rfloor / d\rfloor\) is always satisfied.
Prove that if \(p\) is a prime number and \(1 \leq k \leq p - 1\), then \(\binom{p}{k}\) is divisible by \(p\).
Prove that if \(p\) is a prime number, then \((a + b)^p - a^p - b^p\) is divisible by \(p\) for any integers \(a\) and \(b\).
Prove that there are infinitely many composite numbers among the numbers \(\lfloor 2^k \sqrt{2}\rfloor\) (\(k = 0, 1, \dots\)).
Prove the irrationality of the following numbers:
a) \(\sqrt{3}{17}\)
b) \(\sqrt{2} + \sqrt{3}\)
c) \(\sqrt{2} + \sqrt{3} + \sqrt{5}\)
d) \(\sqrt{3}{3} - \sqrt{2}\)
e) \(\cos 10^{\circ}\)
f) \(\tan 10^{\circ}\)
g) \(\sin 1^{\circ}\)
h) \(\log_{2}3\)
Is it possible for
a) the sum of two rational numbers irrational?
b) the sum of two irrational numbers rational?
c) an irrational number with an irrational degree to be rational?