Wednesday, 31 August 2016

Is there an explicit irrational number which is not known to be either algebraic or transcendental?



There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be either algebraic or transcendental?



Answer



Maybe the best-known example is Apery's constant,
ζ(3)=n=11n3=1.20205,
which Apery proved was irrational a few decades ago; this result is known as Apery's Theorem.



By contrast, ζ(2)=n=11n2 has value π26, which is transcendental because π is.




Apéry, Roger (1979), Irrationalité de ζ(2) et ζ(3), Astérisque (61), 11–13.




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