Transcendental number theory

Transcendental number theory studies numbers that satisfy no nonzero polynomial equation with integer coefficients, and stronger independence properties among numbers and special-function values. The distinction is algebraic: $\sqrt{2}$ is irrational but algebraic because it satisfies $x^2-2=0$.

A useful example is the Liouville constant $L={\sum }_{n\ge 1}10^{-n!}$. Truncating after $N$ terms gives a rational $p_N/q_N$ with $q_N=10^{N!}$ andFor any fixed algebraic degree, these approximations eventually contradict the rational-approximation lower bound in Liouville approximation theorem. Thus exceptionally good approximation can certify transcendence.

Other results, such as the six exponentials theorem, constrain collections of exponential values using linear independence of their inputs. Proving each number irrational is weaker than proving it transcendental, and proving separate transcendence is weaker than proving algebraic independence of a collection.

The Liouville constant is the real numberIts decimal ones occur at factorial positions $1,2,6,24,\dots$. The growing gaps make $L$ exceptionally well approximated by rational numbers. If $p_N/q_N$ is the truncation after $N$ terms, with $q_N=10^{N!}$, thenThe strict lower bound shows the truncations never equal $L$.

For any fixed degree of an algebraic number $d$, these errors eventually fall below every positive constant times $q_N^{-d}$. The Liouville approximation theorem forbids this for an irrational algebraic number of degree $d$. A rational number $a/b$ is also impossible, since distinct fractions satisfy $|a/b-p_N/q_N|\ge 1/(b{q}_N)$. Hence $L$ is a transcendental number. The construction turns deliberately sparse decimal digits into a proof of transcendence.

The six exponentials theorem states that if $x_1,x_2$ are complex numbers linearly independent over $\mathbb{Q}$, and $y_1,y_2,y_3$ are also linearly independent over $\mathbb{Q}$, then at least one of the six numbersis a transcendental number. It does not require the inputs themselves to be algebraic. Both independence hypotheses are essential.^[ref]

For an application, suppose a real $t$ makes $2^t,3^t,5^t$ all algebraic. If $t$ were irrational, choose $x_1=1,x_2=t$ and $y_1=\log 2,y_2=\log 3,y_3=\log 5$. The logarithms are rationally independent: clearing denominators and exponentiating would otherwise give a nontrivial integer relation $2^a{3}^b{5}^c=1$, contradicting unique prime factorization. All six outputs would be algebraic, a contradiction. Hence $t$ must be rational.

With only two independent $y_j$, the analogous assertion is the four exponentials conjecture, not this theorem.