Editing Transcendental number (section)

==Conjectured transcendental numbers==
Numbers which have yet to be proven to be either transcendental or algebraic:
* Most nontrivial combinations of two or more transcendental numbers are themselves not known to be transcendental or even irrational: {{mvar|eπ}}, {{math|''e'' + ''π''}}, {{mvar|π}}<sup>{{mvar|π}}</sup>, {{math|''e''<sup>''e''</sup>}}, {{math|''π''<sup>''e''</sup>}}, {{math|''π''{{sup|{{sqrt|2}}}}}}, {{math|''e''<sup>''π''<sup>2</sup></sup>}}. It has been shown that both {{math|''e'' + ''π''}} and {{math|''π''/''e''}} do not satisfy any [[polynomial equation]] of degree {{math|<math>\leq 8</math>}} and integer coefficients of average size 10<sup>9</sup>.<ref>{{Cite journal |last=Bailey |first=David H. |date=1988 |title=Numerical Results on the Transcendence of Constants Involving $\pi, e$, and Euler's Constant |url=https://www.jstor.org/stable/2007931 |journal=Mathematics of Computation |volume=50 |issue=181 |pages=275–281 |doi=10.2307/2007931 |jstor=2007931 |issn=0025-5718}}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=e |url=https://mathworld.wolfram.com/e.html |access-date=2023-08-12 |website=mathworld.wolfram.com |language=en}}</ref> At least one of the numbers {{math|''e''<sup>''e''</sup>}} and {{math|''e''<sup>''e''<sup>2</sup></sup>}} is transcendental.<ref>{{Cite journal |last=Brownawell |first=W. Dale |date=1974-02-01 |title=The algebraic independence of certain numbers related by the exponential function |journal=Journal of Number Theory |volume=6 |issue=1 |pages=22–31 |doi=10.1016/0022-314X(74)90005-5 |issn=0022-314X|doi-access=free |bibcode=1974JNT.....6...22B }}</ref> [[Schanuel's conjecture]] would imply that all of the above numbers are transcendental and [[Algebraic independence|algebraically independent]].<ref name=":12">{{Cite web |last=Waldschmidt |first=Michel |date=2021 |title=Schanuel's Conjecture: algebraic independence of transcendental numbers |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf}}</ref>
* The [[Euler–Mascheroni constant]] {{mvar|γ}}'':'' In 2010 it has been shown that an infinite list of [[Euler–Lehmer constants|Euler-Lehmer constants]] (which includes {{math|{{var|γ}}/4}}) contains at most one algebraic number.<ref>{{Cite journal |last1=Murty |first1=M. Ram |last2=Saradha |first2=N. |date=2010-12-01 |title=Euler–Lehmer constants and a conjecture of Erdös |journal=[[Journal of Number Theory]] |language=en |volume=130 |issue=12 |pages=2671–2682 |doi=10.1016/j.jnt.2010.07.004 |doi-access=free |issn=0022-314X}}</ref><ref>{{Cite journal |last1=Murty |first1=M. Ram |last2=Zaytseva |first2=Anastasia |date=2013-01-01 |title=Transcendence of generalized Euler constants |journal=[[The American Mathematical Monthly]] |volume=120 |issue=1 |pages=48–54 |doi=10.4169/amer.math.monthly.120.01.048 |s2cid=20495981 |issn=0002-9890}}</ref> In 2012 it was shown that at least one of {{mvar|γ}} and the [[Gompertz constant]] {{mvar|δ}} is transcendental.<ref>{{Cite journal |last=Rivoal |first=Tanguy |date=2012 |title=On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant|journal=Michigan Mathematical Journal |language=en |volume=61 |issue=2 |pages=239–254 |doi=10.1307/mmj/1339011525 |doi-access=free |issn=0026-2285 |url=https://projecteuclid.org/euclid.mmj/1339011525}}</ref>
* The values of the [[Riemann zeta function]] {{math|''ζ''(n)}} at odd positive integers <math>n\geq3</math>; in particular [[Apéry's constant]] {{math|''ζ''(3)}}, which is known to be irrational. For the other numbers {{math|''ζ''(5), ''ζ''(7), ''ζ''(9), ...}} even this is not known. 
* The values of the [[Dirichlet beta function]] {{math|''β''(n)}} at even positive integers <math>n\geq2</math>; in particular [[Catalan's constant|Catalan's Constant]] {{math|''β''(2)}}. (none of them are known to be irrational).<ref>{{Cite journal |last1=Rivoal |first1=T. |last2=Zudilin |first2=W. |date=2003-08-01 |title=Diophantine properties of numbers related to Catalan's constant |url=https://doi.org/10.1007/s00208-003-0420-2 |journal=Mathematische Annalen |language=en |volume=326 |issue=4 |pages=705–721 |doi=10.1007/s00208-003-0420-2 |issn=1432-1807 |s2cid=59328860 |hdl-access=free |hdl=1959.13/803688}}</ref>
* Values of the [[Gamma function|Gamma Function]] {{math|''Γ''(1/n)}} for positive integers <math>n=5</math> and <math>n\geq7</math> are not known to be irrational, let alone transcendental.<ref name=":0">{{cite web |title=Mathematical constants |url=https://www.cambridge.org/us/academic/subjects/mathematics/recreational-mathematics/mathematical-constants |access-date=2022-09-22 |website=Cambridge University Press |language=en |department=Mathematics (general)}}</ref><ref name=":4">{{Cite web |last=Waldschmidt |first=Michel |date=2022 |title=Transcendental Number Theory: recent results and open problems. |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs |website=Michel Waldschmidt}}</ref> For <math>n\geq2</math> at least one the numbers {{math|''Γ''(1/n)}} and {{math|''Γ''(2/n)}} is transcendental.<ref name=":3" />
* Any number given by some kind of [[Limit (mathematics)|limit]] that is not obviously algebraic.<ref name=":4" />