# Gelfond's constant
**Gelfond's constant** is the transcendental number \( e^{\pi} \), the power of [Euler's number](https://www.edgepedia.org/e-mathematical-constant) \( e \) raised to \( \pi \), with numerical value 23.140692632...[1](https://mathworld.wolfram.com/GelfondsConstant.html) • [2](https://proofwiki.org/wiki/Definition%3AGelfond%27s_Constant) Its transcendence follows from the Gelfond–Schneider theorem, proved by Aleksandr Osipovich Gelfond in 1934 and independently by [Theodor Schneider](https://www.edgepedia.org/theodor-schneider) in 1935, which settled the seventh of [David Hilbert](https://www.edgepedia.org/david-hilbert)'s 23 problems announced in 1900.[3](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E) The name is not universally fixed; Wolfram MathWorld notes that the constant "seems to lack a generally accepted name" and adopts "Gelfond's constant" for it.[1](https://mathworld.wolfram.com/GelfondsConstant.html)
| Key fact | Detail |
|---|---|
| Definition and value | \( e^{\pi} = 23.140692632\ldots \) (OEIS A039661)[1](https://mathworld.wolfram.com/GelfondsConstant.html) |
| Transcendence | A value of \( i^{-2i} = \exp\{-2i \log i\} \) is \( e^{\pi} \), so the Gelfond–Schneider theorem makes it transcendental[3](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E) |
| Historical role | Singled out, with \( 2^{\sqrt{2}} \), in Hilbert's seventh problem (1900) as a number whose transcendence was open[1](https://mathworld.wolfram.com/GelfondsConstant.html) |
| Proof dates | Gelfond published in 1934; Schneider gave an independent proof in 1935[3](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E) |
| Continued fraction | \( [23, 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, \ldots] \) (OEIS A058287)[1](https://mathworld.wolfram.com/GelfondsConstant.html) |
| Related almost-integer | \( e^{\pi\sqrt{163}} = (e^{\pi})^{\sqrt{163}} \), Ramanujan's constant, equals 262537412640768743.99999999999925007...[4](https://oeis.org/wiki/Ramanujan%27s_constant) |
| Fast computation | An AGM-type iteration with \( k_0 = 1/\sqrt{2} \) yields \( e^{\pi} \approx (1/(4k_n))^{-2^{1-n}} \)[1](https://mathworld.wolfram.com/GelfondsConstant.html) |
## Hilbert's seventh problem and the proof of transcendence
In 1900 David Hilbert announced a list of twenty-three outstanding unsolved problems at the International Congress of Mathematicians; the seventh asked about the irrationality and transcendence of certain numbers, specifically powers \( \alpha^{\beta} \) with algebraic bases and exponents.[3](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E) • [5](https://ideas.repec.org/h/spr/sprchp/978-981-15-4155-1_3.html) Both \( e^{\pi} \) and the Gelfond–Schneider constant \( 2^{\sqrt{2}} \) were singled out in this problem as examples whose transcendence was open.[1](https://mathworld.wolfram.com/GelfondsConstant.html) Partial progress came in 1929 from Gelfond and Kuzmin; the complete solution was published by Gelfond in 1934, with Schneider's independent proof following in 1935.[3](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E) • [6](https://www.ams.org/books/surv/019/surv019-endmatter.pdf)
The theorem states: if \( \alpha \) and \( \beta \) are algebraic numbers with \( \alpha \neq 0 \), \( \alpha \neq 1 \), and \( \beta \) is not a real rational number, then any value of \( \alpha^{\beta} \) is transcendental.[3](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E) The phrase "any value" matters because \( \alpha^{\beta} = \exp\{\beta \log \alpha\} \) is in general multivalued. Applying the theorem to \( \alpha = i \) and \( \beta = -2i \), one value of \( i^{-2i} = \exp\{-2i \log i\} \) is \( e^{\pi} \), so \( e^{\pi} \) is transcendental.[3](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E)
That same paper also established a more general result: working separately, Gelfond and Schneider showed that when the base of a logarithm is algebraic and neither 0 nor 1, the logarithm of any algebraic number taken to that base must be either rational or transcendental.[7](https://www.cambridge.org/core/journals/mathematika/article/linear-forms-in-the-logarithms-of-algebraic-numbers/55674A9E93007ADD34DCEDC12EA4FB4D) Gelfond later remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be valuable for difficult problems of number theory, a direction Alan Baker developed in his 1966 work on linear forms.[7](https://www.cambridge.org/core/journals/mathematika/article/linear-forms-in-the-logarithms-of-algebraic-numbers/55674A9E93007ADD34DCEDC12EA4FB4D)
## Numerical character
The decimal expansion begins \( e^{\pi} = 23.140692632\ldots \), and its simple continued fraction begins \( [23, 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, \ldots] \); both sequences are cataloged in the [On-Line Encyclopedia of Integer Sequences](https://www.edgepedia.org/on-line-encyclopedia-of-integer-sequences) as A039661 and A058287.[1](https://mathworld.wolfram.com/GelfondsConstant.html) The large partial quotient 591 appears early in the continued fraction, which is the kind of feature that makes a constant's continued fraction expansion of independent interest.[1](https://mathworld.wolfram.com/GelfondsConstant.html)
**Fast digit computation.** The digits can be computed efficiently using the iteration
\[ k_n = \frac{1 - \sqrt{1 - k_{n-1}^2}}{1 + \sqrt{1 - k_{n-1}^2}}, \qquad k_0 = \frac{1}{\sqrt{2}}, \]
and then plugging into
\[ e^{\pi} \approx \left( \frac{1}{4 k_n} \right)^{-2^{1-n}} \]
(Borwein and Bailey, 2003, p. 137).[1](https://mathworld.wolfram.com/GelfondsConstant.html) This belongs to the family of quadratically convergent algorithms built on the Gaussian arithmetic-geometric mean, which [Jonathan Borwein](https://www.edgepedia.org/jonathan-borwein) and [Peter Borwein](https://www.edgepedia.org/peter-borwein), mathematicians at [Simon Fraser University](https://www.edgepedia.org/simon-fraser-university) and Dalhousie University respectively, analyzed systematically for fast computation of \( \pi \) and elementary functions.[8](https://epubs.siam.org/doi/10.1137/1026073) Multiple-precision libraries have also computed \( e^{\pi} \) from a representation of the value by an infinite product, with guaranteed interval enclosures.[9](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0076539208628519)
**Hypergeometric forms.** A 2020 note by A. K. Rathie and R. B. Paris records representations of the constant through Gauss hypergeometric functions, including
\[ e^{\pi} = {}_2F_1\!\left(i, -i; \tfrac{1}{2}; 1\right) + 2\, {}_2F_1\!\left(\tfrac{1}{2}+i, \tfrac{1}{2}-i; \tfrac{3}{2}; 1\right), \]
established via the classical Gauss summation theorem, and develops natural \( {}_3F_2 \) extensions of the constant and its square root.[10](https://arxiv.org/abs/2007.10161)
## Connections to other constants
Gelfond's constant generates further transcendental numbers by exponentiation: since \( \sqrt{163} \) is algebraic and irrational, \( (e^{\pi})^{\sqrt{163}} = e^{\pi\sqrt{163}} \) is transcendental as well.[4](https://oeis.org/wiki/Ramanujan%27s_constant) This number, known as Ramanujan's constant, is famous as an [almost integer](https://www.edgepedia.org/almost-integer):
\[ e^{\pi\sqrt{163}} = 262537412640768743.99999999999925007\ldots \]
The constant also appears in analysis outside number theory. A geometrical occurrence arises in the sum of even-dimension unit spheres, whose volumes are \( V_{2n} = \pi^n / n! \).[10](https://arxiv.org/abs/2007.10161) Approximations to the Feigenbaum constant, the universal constant of period-doubling bifurcations, have been expressed using \( e^{\pi} \), giving about six correct digits to the right of the decimal point.[11](https://mathworld.wolfram.com/FeigenbaumConstantApproximations.html)
## What the theorem does not cover
The Gelfond–Schneider theorem applies when the exponent is algebraic and irrational. Within its scope, however, the theorem established a large class of transcendental numbers at once, and the Gelfond–Schneider method became the main instrument for studying algebraic independence of values of functions satisfying algebraic differential and functional equations.[6](https://www.ams.org/books/surv/019/surv019-endmatter.pdf) The theorem has even been machine-verified: Michail Karatarakis and Freek Wiedijk, researchers at Radboud University's Institute for Computing and Information Sciences, formalized Hilbert's seventh problem and its solution in the Lean 4 proof assistant.[12](https://arxiv.org/abs/2603.24823)
## References
1. [Weisstein, Eric W. "Gelfond's Constant." MathWorld–A Wolfram Resource.](https://mathworld.wolfram.com/GelfondsConstant.html)
2. [Definition:Gelfond's Constant, ProofWiki](https://proofwiki.org/wiki/Definition%3AGelfond%27s_Constant)
3. [Ivan Niven, "The Gelfond–Schneider Theorem," Chapter X of *Irrational Numbers*, Mathematical Association of America, 1956, pp. 134–150.](https://www.cambridge.org/core/books/abs/irrational-numbers/gelfondschneider-theorem/BEB195D155B5E21FC90467893CFBEE6E)
4. [Ramanujan's constant, OeisWiki](https://oeis.org/wiki/Ramanujan%27s_constant)
5. ["Theorem of Gelfond and Schneider," in *Pillars of Transcendental Number Theory*, Springer, chapter 0, pp. 35–44.](https://ideas.repec.org/h/spr/sprchp/978-981-15-4155-1_3.html)
6. [Preface, Mathematical Surveys 019, American Mathematical Society.](https://www.ams.org/books/surv/019/surv019-endmatter.pdf)
7. [Alan Baker, "Linear forms in the logarithms of algebraic numbers," *Mathematika* 13 (1966), pp. 204–216.](https://www.cambridge.org/core/journals/mathematika/article/linear-forms-in-the-logarithms-of-algebraic-numbers/55674A9E93007ADD34DCEDC12EA4FB4D)
8. [J. M. Borwein and P. B. Borwein, "The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions," *SIAM Review* 26 (1984), pp. 351–366.](https://epubs.siam.org/doi/10.1137/1026073)
9. [Multiple-Precision Computations with Result Verification, Mathematics in Science and Engineering, Volume 189, 1993, pp. 325–356.](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0076539208628519)
10. [A. K. Rathie and R. B. Paris, "A note on an extension of Gelfond's constant," arXiv:2007.10161 (2020).](https://arxiv.org/abs/2007.10161)
11. [Feigenbaum Constant Approximations, MathWorld](https://mathworld.wolfram.com/FeigenbaumConstantApproximations.html)
12. [Michail Karatarakis and Freek Wiedijk, "A formalization of the Gelfond–Schneider theorem," arXiv:2603.24823.](https://arxiv.org/abs/2603.24823)
---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Special and named integers › Almost integers*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license