Show that any natural number \(n\) can be uniquely represented in the form \(n = \binom{x}{1} + \binom{y}{2} + \binom{z}{3}\) where \(x, y, z\) are integers such that \(0 \leq x < y < z\), or \(0 = x = y < z\).
Here is a fragment of the table, which is called the Leibniz triangle. Its properties are “analogous in the sense of the opposite” to the properties of Pascal’s triangle. The numbers on the boundary of the triangle are the inverses of consecutive natural numbers. Each number is equal to the sum of two numbers below it. Find the formula that connects the numbers from Pascal’s and Leibniz triangles.
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 \(d >30,000\).
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 in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.
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) \(\log_{2}3\)
One of the roots of the equation \(x^2 + ax + b = 0\) is \(1 + \sqrt 3\). Find \(a\) and \(b\) if you know that they are rational.