The Gelfond–Schneider constant or Hilbert number is
two to the
power of the
square root of two
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. T ...
:
:2 ≈ ...
which was proved to be a
transcendental number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
by
Rodion Kuzmin in 1930.
In 1934,
Aleksandr Gelfond and
Theodor Schneider independently proved the more general ''
Gelfond–Schneider theorem'', which solved the part of
Hilbert's seventh problem described below.
Properties
The
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of the Gelfond–Schneider constant is the transcendental number
:
....
This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either
is a rational which proves the theorem, or it is irrational (as it turns out to be) and then
:
is an irrational to an irrational power that is a rational which proves the theorem. The proof is not
constructive, as it does not say which of the two cases is true, but it is much simpler than
Kuzmin's proof.
Hilbert's seventh problem
Part of the seventh of
Hilbert's twenty-three problems posed in 1900 was to prove, or find a counterexample to, the claim that ''a
b'' is always transcendental for algebraic ''a'' ≠ 0, 1 and irrational algebraic ''b''. In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2
.
In 1919, he gave a lecture on
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and spoke of three conjectures: the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
,
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
, and the transcendence of 2
. He mentioned to the audience that he didn't expect anyone in the hall to live long enough to see a proof of this result.
[David Hilbert, ''Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920''.] But the proof of this number's transcendence was published by Kuzmin in 1930,
[ well within ]Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad ...
's own lifetime. Namely, Kuzmin proved the case where the exponent ''b'' is a real quadratic irrational
In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numb ...
, which was later extended to an arbitrary algebraic irrational ''b'' by Gelfond and by Schneider.
See also
* Gelfond's constant
References
Further reading
*
*
{{DEFAULTSORT:Gelfond-Schneider Constant
Real transcendental numbers
Mathematical constants