A list of the statements appearing in our posts about number theory follows.
Title and link  Statement 

Theorem N.1: Fundamental theory of arithmetic (not proved)  Every integer number greater than 1 can be written as a product of prime numbers. Moreover such expression, called factorization or decomposition into prime factors, is unique apart from the order of the factors. 
Theorem N.2: Infinity of primes  The number of primes is infinite. 
Property N.1: Overestimation of binomials  For all n > 0 and for all m: \binom{n}{m} \leq 2^{n1} 
Property N.2: Underestimation of the central binomials of the Pascal’s triangle  For all n \geq 2: \binom{n}{\frac{n}{2}} \geq \frac{2^n}{n} if n is even, and \binom{n}{\left \lceil \frac{n}{2} \right \rceil} = \binom{n}{\left \lfloor \frac{n}{2} \right \rfloor} \geq \frac{2^n}{n} if n is odd. 
Property N.3: Underestimation of the central binomial of Pascal’s triangle, even n  For all even n \geq 2: \binom{n}{\frac{n}{2}} \geq \sqrt{2^{n}} 
Proposition N.1: Majorization of the product of primes up to x  For all x > 0: \theta^{\star}(x) = \prod_{1 \leq p \leq x} p \leq 2^{2(x1)} 
Theorem N.3: Bertrand’s postulate  For all n > 0, there exists a prime number between n + 1 and 2n. 
Lemma N.1: Factorization of the binomial \binom{2n}{n}  In the factorization of the binomial \binom{2n}{n}, for n > 0, a prime number p cannot have an exponent greater than \log_p 2n. 
Proposition N.2: Computation of \psi^{\star}(x)  For every integer x > 0: \psi^{\star}(x) = \prod_{p \leq x} p^{\left \lfloor \log_p x \right \rfloor} 
Corollary of Proposition N.2: Overestimation of \psi^{\star}(x) with \pi(x)  For every integer x > 0: \psi^{\star}(x) \leq x^{\pi(x)} 
Proposition N.3: Underestimation of \psi^{\star}(x)  For every integer x > 0: \psi^{\star}(x) \geq \sqrt[3]{2^x} 
Lemma N.2: Reformulation of \psi^{\star}(x) (“calculation by rows”)  For every integer x > 0: \psi^{\star}(x) = \left(\prod_{p \leq x} p\right) \cdot \left(\prod_{p^2 \leq x} p\right) \cdot \dots \cdot \left(\prod_{p^R \leq x} p\right) 
Proposition N.4: Connection between \psi^{\star} and \theta^{\star} functions  For every integer x > 0: \psi^{\star}(x) = \theta^{\star}(x) \cdot \theta^{\star}(\sqrt{x}) \cdot \dots \cdot \theta^{\star}(\sqrt[R]{x}), where R := \left \lfloor \log_2 x \right \rfloor. 
Lemma N.3: Underestimation of \theta^{\star}(x) through \pi(x)  For every real number \delta \geq 0 and for every x \gt 1: \theta^{\star}(x) \gt \left(x^{\delta}\right)^{\pi(x)  x^{\delta}}. 
Theorem N.4: Asymptotic equivalence and order of magnitude of \theta(x) and \psi(x)  The functions \theta(x) and \psi(x) are asymptotically equivalent and have order x: \theta(x) \sim \psi(x), \theta(x) \asymp x \asymp \psi(x). 
Theorem N.5: Asymptotical equivalence between \pi(x) and \frac{\theta(x)}{\log x}  \pi(x) \sim \frac{\theta(x)}{\log x} 
Corollary of Theorem N.5: Asymptotical equivalence between \pi(x) and \frac{\psi(x)}{\log x}  \pi(x) \sim \frac{\psi(x)}{\log x} 
Corollary II of Theorem N.5: Chebyshev’s Theorem: order of magnitude of \pi(x)  \pi(x) \asymp \frac{x}{\log x} 
Lemma N.4: Lemma of bar chart area  Let c_1, c_2, \dots, c_n be natural numbers, with n > 0. Let f: \{1, 2, ..., n\} \rightarrow \mathbb{R} be a function. Then the area A of the bar chart made up of n rectangles, each having basis c_i and height f(i), given by A = c_1 f(1) + c_2 f(2) + \ldots + c_n f(n) = \sum_{i=1}^{n} c_i f(i), can also be computed with the formula \begin{aligned}A &= C_n f(n) + C_{n1} (f(n1)  f(n)) + \ldots + C_1 (f(1)  f(2)) \\&= \sum_{k = 1}^{n1} C_k (f(k)  f(k + 1)) + C_n f(n)\end{aligned} where C_k := c_1 + c_2 + \ldots + c_k = \sum_{i=1}^{k} c_i. 
Lemma N.5: Lemma of bar chart area, second form  Let c_1, c_2, \dots, c_n be natural numbers, with n > 0. Let f: \{1, 2, ..., n\} \rightarrow \mathbb{R} be a function, and \widetilde{f} a C^1 extension of f. Then the area A of the bar chart made up of n rectangles, each with basis c_i and height f(i), given by A = c_1 f(1) + c_2 f(2) + \ldots + c_n f(n) = \sum_{i=1}^{n} c_i f(i), can be also computed with the formula C(n) f(n)  \int_1^n \overline{C}(k) \widetilde{f}'(t) dt where C: \{1, 2, ..., n\} \rightarrow \mathbb{N} is the function defined by C(k) := c_1 + c_2 + \ldots + c_k = \sum_{i=1}^{k} c_i. 
Theorem N.6: Approximation of the sum of inverses of the first positive integers 
For all integers n \gt 0: \sum_{k = 1}^n \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} \approx \log n + \gamma
where logarithm base is Napier’s number e, while \gamma \approx 0.58 is the Euler’s constant. In particular: \sum_{k = 1}^n \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} = \log n + \gamma + O\left(\frac{1}{n}\right)

Property N.4: Upper and lower bounds for the value assumed by a function defined on integer numbers, by means of integrals of an extension 
Let f: I \rightarrow \mathbb{R} be a function defined on a set I \subset \mathbb{Z}.
Let \underset{\sim}{f}: \underline{I} \rightarrow \mathbb{R} be an extension of f, where \underline{I} := \bigcup_{n \in I} (n  1, n]. Then:

Lemma N.6: Order of magnitude of the sum of a fraction logarithms when the denominator changes  \sum_{n=1}^x \log \left(\frac{x}{n} \right) = O(x). 