Classical Constants Generated from Prime Numbers: A Pedagogical Bridge Between the Discrete and the Continuous

1. Introduction

In the teaching of calculus and real analysis, the constants

$$e,\pi ,\phi ={\displaystyle \frac{1+\sqrt{5}}{2}}$$

are typically introduced via power series, limits of elementary functions, or geometric interpretations. On the other hand, in elementary number theory courses one introduces the set of prime numbers

$$2,3,5,7,11,13,\dots ,$$

together with functions such as $\pi \left(x\right)$ (the prime counting function) and sequences such as the Fibonacci sequence.

The central message of this work is that these two apparently separate narratives can be conceptually connected in a very appealing way: it is possible to define or approximate classical constants using only prime numbers, either in the form of limits, infinite products, or sequences indexed by primes. Although some of these constructions are not competitive as numerical methods, they are very valuable as pedagogical tools to illustrate the bridge between the discrete (primes) and the continuous (limits, derivatives, integrals).

In the following sections we present:

• limits that generate e from the distribution of primes,
• products over primes that lead to $\pi$,
• the golden ratio $\phi$ obtained through Fibonacci and Lucas numbers evaluated at prime indices,
• and a pedagogical discussion with suggested activities.

Standard references for analytic number theory and the distribution of primes, including the Prime Number Theorem and Euler products, are Apostol [1] and Hardy–Wright [2]; Finch’s monograph [3] surveys many classical constants and their representations from a unified perspective.

2. Preliminaries: Primes and Classical Functions

We denote by $p_n$ the n-th prime number and by $\pi \left(x\right)$ the prime-counting function:

$$\pi \left(x\right)=\#\{p\mathrm{prime}:p\le x\}.$$

The Prime Number Theorem states that

$$\pi \left(x\right)\sim {\displaystyle \frac{x}{logx}}\mathrm{as}x\to \infty ,$$

and, equivalently,

$$p_n\sim nlogn\mathrm{as}n\to \infty .$$

We do not use deep technical versions of this result; we only need its most basic asymptotic form, which is enough to justify the limits that appear in the following sections (see, for instance, [1,4]).

We also introduce the Chebyshev functions

$$\theta \left(x\right)=\sum _{p\le x}logp,\psi \left(x\right)=\sum _{p^k\le x}logp.$$

For the pedagogical purposes of this article it suffices to know that $\theta \left(x\right)\sim x$ as $x\to \infty$, and that the identity

$$\zeta \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^s}}=\prod _p{\displaystyle \frac{1}{1-p^{-s}}}$$

(for $\Re \left(s\right)>1$) links primes directly to series over the integers.

3. The Constant e from Prime Numbers

3.1. A Conceptual Construction: $n^{n/p_n}\to e$

Let $p_n$ denote the n-th prime. Consider the sequence

$$a_n=n^{n/p_n}.$$

Taking logarithms, we obtain

$$log{a}_n={\displaystyle \frac{n}{p_n}}logn.$$

By the Prime Number Theorem, $p_n\sim nlogn$, so that

$${\displaystyle \frac{n}{p_n}}logn\sim {\displaystyle \frac{n}{nlogn}}logn=1.$$

Therefore $log{a}_n\to 1$ and, consequently:

Proposition 1. 

We have

$$\underset{n\to \infty }{lim}{n}^{n/p_n}=e.$$

This construction is conceptually elegant: it defines e as the limit of an expression that involves only integers and the positions of the primes. However, the convergence is extremely slow. Table 1 shows some illustrative values, where we can see that even for $n={10}^5$ the value is far from e.

Here $e\approx 2.7182818\dots$. The table is useful to discuss in class the difference between existence of a limit and speed of convergence.

3.2. A Product Construction with Faster Decay

Proposition 1 shows that e is “encoded” in the distribution of primes. However, it is natural to look for expressions that mimic the structure of products over primes that appear in $\zeta \left(2\right)$ and that also converge relatively fast.

Let

$$S=\sum _p{\displaystyle \frac{1}{p^2}},$$

which is convergent because it is a subseries of ${\sum }_{n\ge 1}1/n^2$. We define, purely algebraically,

$$e=exp\left(1\right)=exp\left(\sum _p{\displaystyle \frac{1}{S{p}^2}}\right).$$

Rewriting this as a product, we obtain

$$e=\prod _{p}exp\left({\displaystyle \frac{1}{S{p}^2}}\right).$$

This suggests the sequence of partial products

$$E_N=\prod _{p\le p_N}exp\left({\displaystyle \frac{1}{S{p}^2}}\right),$$

where $p_N$ is the N-th prime. Then we have:

Proposition 2. 

With the above notation,

$$\underset{N\to \infty }{lim}{E}_N=e.$$

Moreover, the error in the exponent behaves, at first order, like

$$\left|\sum _{p>p_N}{\displaystyle \frac{1}{S{p}^2}}\right|\lesssim {\displaystyle \frac{C}{p_N}}$$

for some constant $C>0$, so that the convergence is of order $1/p_N$.

Table 2 shows some typical values (assuming $S\approx 0.4522$), where we can observe a faster convergence than in Table 1.

These tables allow us to compare, in the classroom, two different ways of “extracting” the same constant e from prime arithmetic.

4. The Constant $\pi$ from Products over Primes

The case of $\pi$ is more classical and is based on the value of the Riemann zeta function at $s=2$:

$$\zeta \left(2\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^2}}={\displaystyle \frac{{\pi }^2}{6}}.$$

On the other hand, Euler’s product formula gives (for $\Re \left(s\right)>1$)

$$\zeta \left(s\right)=\prod _p{\displaystyle \frac{1}{1-p^{-s}}},$$

so that at $s=2$ we obtain

$${\displaystyle \frac{{\pi }^2}{6}}=\prod _p{\displaystyle \frac{1}{1-\frac{1}{p^2}}}.$$

This allows us to write:

Proposition 3. 

We have

$$\pi =\sqrt{6{\prod }_p{\displaystyle \frac{1}{1-\frac{1}{p^2}}}}=\underset{x\to \infty }{lim}\sqrt{6{\prod }_{p\le x}{\displaystyle \frac{1}{1-\frac{1}{p^2}}}}.$$

If we define the partial product

$$\pi \left(x\right)=\sqrt{6{\prod }_{p\le x}{\displaystyle \frac{1}{1-\frac{1}{p^2}}}}$$

we obtain a monotonically increasing approximation to $\pi$. Table 3 illustrates some values for the first few primes.

Although the convergence is not immediate either, it is significantly better than in Table 1, and the comparison is quite instructive for students.

5. The Golden Ratio via Primes, Fibonacci, and Lucas

The Fibonacci sequence ${\left(F_n\right)}_{n\ge 0}$ (see, e.g., [5]) is defined by

$$F_0=0,F_1=1,F_{n+1}=F_n+F_{n-1},$$

and the golden ratio is the limit

$$\phi =\underset{n\to \infty }{lim}{\displaystyle \frac{F_{n+1}}{F_n}}.$$

Binet’s formula gives

$$F_n={\displaystyle \frac{{\phi }^n-{\psi }^n}{\sqrt{5}}},\mathrm{where}\psi =-{\displaystyle \frac{1}{\phi }}.$$

In particular, $F_n\sim {\phi }^n/\sqrt{5}$.

Let $p_n$ be the n-th prime. We can use primes as a sampling tool:

Proposition 4. 

The following limits hold:

$$\phi =\underset{n\to \infty }{lim}{\displaystyle \frac{F_{p_n+1}}{F_{p_n}}}and\phi =\underset{n\to \infty }{lim}{\left(F_{p_n}\right)}^{1/p_n}.$$

(Idea of the proof). 

Since $p_n\to \infty$ as $n\to \infty$, we simply note that

$${\displaystyle \frac{F_{p_n+1}}{F_{p_n}}}\to \phi$$

is just the same convergence as $F_{n+1}/F_n\to \phi$ restricted to a subsequence (any subsequence of a convergent sequence converges to the same limit). For the second identity, from Binet’s formula we have

$$F_{p_n}\sim {\displaystyle \frac{{\phi }^{p_n}}{\sqrt{5}}},$$

so that

$${\left(F_{p_n}\right)}^{1/p_n}\sim {\left({\displaystyle \frac{{\phi }^{p_n}}{\sqrt{5}}}\right)}^{1/p_n}=\phi {\left({\displaystyle \frac{1}{\sqrt{5}}}\right)}^{1/p_n}\to \phi .$$

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An analogous construction can be carried out with the Lucas sequence ${\left(L_n\right)}_{n\ge 0}$, defined by $L_0=2$, $L_1=1$, $L_{n+1}=L_n+L_{n-1}$, which satisfies $L_n\sim {\phi }^n$ (see also [5]):

Proposition 5. 

Let $p_n$ be the n-th prime. Then

$$\phi =\underset{n\to \infty }{lim}{\displaystyle \frac{L_{p_n+1}}{L_{p_n}}}and\phi =\underset{n\to \infty }{lim}{\left(L_{p_n}\right)}^{1/p_n}.$$

In these constructions, primes do not directly generate a product or a series, but they act as a selection tool: we keep only the Fibonacci or Lucas terms at prime indices and, nevertheless, the golden ratio reappears as a limit.

5.1. Combining $\pi$ and $\phi$ with Products over Primes

A different approach, closer in spirit to Proposition 3, is to use the identity

$$\phi =2cos\left({\displaystyle \frac{\pi }{5}}\right).$$

If we define $\pi \left(x\right)$ as in Proposition 3, that is,

$$\pi \left(x\right)=\sqrt{6{\prod }_{p\le x}{\displaystyle \frac{1}{1-\frac{1}{p^2}}}}$$

then

$$\phi \left(x\right)=2cos\left({\displaystyle \frac{\pi \left(x\right)}{5}}\right)⟶2cos\left({\displaystyle \frac{\pi }{5}}\right)=\phi (x\to \infty ).$$

In compact notation:

Proposition 6. 

We have

$$\phi =\underset{x\to \infty }{lim}2cos\left({\displaystyle \frac{1}{5}}\sqrt{6{\prod }_{p\le x}{\displaystyle \frac{1}{1-\frac{1}{p^2}}}}\right).$$

Table 4 illustrates some values of $\phi \left(x\right)$ obtained by truncating the product in $\pi \left(x\right)$ at the first primes.

This example combines three didactic ideas:

• products over primes,
• trigonometry and continuous functions,
• and the golden ratio as a geometric constant.

6. Pedagogical Discussion and Suggested Activities

In this section we outline some ways to use the previous constructions in the classroom.

6.1. Comparing Rates of Convergence

A first activity is to ask students to compare numerically different expressions for the same constant. For example, for e:

• the classical definition $e={lim}_{n\to \infty }{\left(1+{\displaystyle \frac{1}{n}}\right)}^n$;
• the limit $e={lim}_{n\to \infty }{n}^{n/p_n}$;
• the partial product $E_N={\prod }_{p\le p_N}exp\left({\displaystyle \frac{1}{S{p}^2}}\right)$.

Using Table 1 and Table 2 one can discuss:

• how fast each expression approaches e,
• how many operations each method requires,
• what information about primes is needed to evaluate them.

A second activity consists in doing the same for $\pi$ and $\phi$:

• classical trigonometric formulas for $\pi$,
• the approximation via the prime product $\pi \left(x\right)=\sqrt{6{\prod }_{p\le x}{(1-p^{-2})}^{-1}}$ (Table 3),
• and, from this, the approximation to $\phi$ via $\phi \left(x\right)=2cos\left(\pi \right(x)/5)$ (Table 4).

6.2. A Bridge Between Number Theory and Analysis

Another interesting pedagogical angle is to use these examples to tell a unified story:

• Primes appear as the fundamental building blocks of arithmetic.
• The zeta function, Euler products, and the Prime Number Theorem show how they are related to series and limits.
• Analytical constants such as e, $\pi$, and $\phi$ can be viewed as continuous shadows of discrete structures.

Depending on the level of the course, one can adjust the rigor:

• In a calculus course, Proposition 1 can be presented as a simple application of the limit $log{a}_n\to 1$.
• In a real analysis or number theory course, one can explore in more depth the ideas behind the Prime Number Theorem and Euler’s product (see [1,2,4]).

6.3. Open Explorations

Finally, these constructions can lead to small guided “research projects”. For example:

• Define new constants by combining products over primes (for instance, inspired by the twin prime constant) and study their numerical convergence.
• Investigate what happens if one replaces the sequence of all natural numbers by a sequence indexed only by primes in other known limits.
• Explore variants of Propositions 4 and 5 by restricting the indices to special subsets of primes (for example, twin primes, Sophie Germain primes, etc.).

7. Conclusions

We have shown that it is possible to construct classical constants such as e, $\pi$, and $\phi$ from expressions involving only primes, either via limits, products, or sequences indexed by primes. Although some of these formulations are not efficient for numerical computation, they provide fertile ground for pedagogical activities connecting number theory and analysis.

In particular, the proposed viewpoint can help to:

• visualize the Prime Number Theorem as a description of the rate at which certain limits approach e,
• reinterpret Euler’s product for $\zeta \left(2\right)$ as a machine that generates $\pi$ from primes,
• and see the golden ratio as a robust limit that persists even when we sample the Fibonacci and Lucas sequences only at prime indices.

We believe that these ideas can serve as the basis for didactic material, lecture notes, or expository articles that motivate students by showing the deep interplay between discrete objects (primes, recursive sequences) and analytic constants, in the spirit of [3,5].