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A more recent proof by Wadim Zudilin is more reminiscent of Apéry's original proof, [6] and also has similarities to a fourth proof by Yuri Nesterenko. [7] These later proofs again derive a contradiction from the assumption that ζ ( 3 ) {\displaystyle \zeta (3)} is rational by constructing sequences that tend to zero but are bounded below by ...
In 1840, Liouville published a proof of the fact that e 2 is irrational [10] followed by a proof that e 2 is not a root of a second-degree polynomial with rational coefficients. [11] This last fact implies that e 4 is irrational. His proofs are similar to Fourier's proof of the irrationality of e.
Transcendental numbers therefore represent the typical case; even so, it may be extremely difficult to prove that a given number is transcendental (or even simply irrational). For this reason transcendence theory often works towards a more quantitative approach.
Like how 3+5 is the only way to break 8 into two primes, but 42 can broken into 5+37, 11+31, 13+29, and 19+23. So it feels like Goldbach’s Conjecture is an understatement for very large numbers.
Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants ζ(2n + 1) are irrational. [7] In particular at least one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational. [8] Apéry's constant has not yet been proved transcendental, but it is known to be an algebraic period ...
He has reproved Apéry's theorem that ζ(3) is irrational, and expanded it. Zudilin proved that at least one of the four numbers ζ(5), ζ(7), ζ(9), or ζ(11) is irrational. [ 2 ] For that accomplishment, he won the Distinguished Award of the Hardy - Ramanujan Society in 2001.
A more general proof shows that the mth root of an integer N is irrational, unless N is the mth power of an integer n. [7] That is, it is impossible to express the m th root of an integer N as the ratio a ⁄ b of two integers a and b , that share no common prime factor , except in cases in which b = 1.
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