Opinion – Marcelo Viana: The three most famous constants in mathematics

Earlier, I dealt with four major mathematical problems here:

1. The twin cousins ​​conjecture
2. Goldbach’s conjecture
3. Collatz conjecture
4. Riemann’s Hypothesis

I continue the list with two problems relating to the three most famous constants in mathematics. They have been studied since the 18th century, but have not yet been resolved.

5. Is the number π+e rational? There is no doubt that π (pronounce pi) =3.14159… is the most famous constant in mathematics, closely followed by the Euler-Neper number e=2.71828…

We know that both are irrational numbers, that is, they cannot be written as p/q fractions of whole numbers. This was proved by Leonhard Euler in 1737, in the case of e, and by Johann Heinrich Lambert, around 1760, in the case of π.

In fact, we know more: Ferdinand von Lindemann proved in 1882 that both π and e are transcendental numbers, that is, they are not solutions of any polynomial equation akx^k+ … +a2x^two+a1x+a0=0 with coefficients ak, …, a2, a1, a0 integers.

But for most numbers constructed from them, such as π+e, π-e, πe, π/e, π^πand^and and π^and, we have no idea whether they are rational or irrational. A somewhat surprising exception is and^πwhich is known to be transcendent, hence irrational.

6. Is the number γ rational? In the most famous race for mathematical constant, the bronze medal goes to the Euler-Mascheroni number γ (pronounce gamma) =0.57721… It appeared in works by Leonhard Euler in 1734 and by Lorenzo Mascheroni in 1790.

The definition is as follows: add the fractions 1/1, 1/2, 1/3, … up to 1/N and subtract the value of the Naperian logarithm from N; the larger the N, the closer the result will be to the value of γ. The number γ has been studied a lot, and we know that it is related to important questions in different areas of mathematics. We also know over 600 billion of its digits.

Everyone believes that γ is irrational, but there is no rigorous demonstration of this fact. For example, in 2010 mathematicians M. Ram Murty and N. Saradha found a certain infinite family of numbers containing γ, and proved that at most one of them can be rational. We don’t know which one, and it would be too much of a coincidence that it was just γ, do you agree? But we also can’t guarantee that it won’t be…