Baker theorem on linear forms in logarithms
E637308
The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
All labels observed (5)
How this entity was disambiguated
This entity first appeared as the object of triple T7030803 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Baker theorem on linear forms in logarithms Context triple: [Diophantine approximation, hasKeyResult, Baker theorem on linear forms in logarithms]
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A.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
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B.
Mertens’ theorems
Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.
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C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
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D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
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E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Baker theorem on linear forms in logarithms Target entity description: The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
-
A.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
-
B.
Mertens’ theorems
Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
E.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in transcendental number theory ⓘ |
| appearsIn |
Alan Baker’s work on transcendental number theory
ⓘ
monographs on Diophantine approximation ⓘ textbooks on transcendental number theory ⓘ |
| appliesTo | nonzero linear combinations of logarithms of algebraic numbers ⓘ |
| characterizedBy | effectivity of the bounds obtained ⓘ |
| concerns |
explicit lower bounds for linear combinations of logarithms
ⓘ
linear forms in logarithms of algebraic numbers ⓘ |
| contributedTo | Alan Baker receiving the Fields Medal in 1970 ⓘ |
| field |
Diophantine approximation
ⓘ
Diophantine equations ⓘ number theory ⓘ transcendental number theory ⓘ |
| generalizes | earlier results of Gelfond and Schneider ⓘ |
| hasConsequence |
bounds for exponents in exponential Diophantine equations can be made explicit
ⓘ
effective irrationality measures for certain algebraic numbers ⓘ effective lower bounds for linear forms in logarithms of algebraic numbers ⓘ many Diophantine equations have only finitely many integer solutions ⓘ |
| implies | linear forms in logarithms of algebraic numbers are rarely very small ⓘ |
| involves |
algebraic number fields
ⓘ
degree of algebraic numbers ⓘ explicit constants depending on degrees and heights ⓘ heights of algebraic numbers ⓘ logarithms on the complex plane ⓘ |
| namedAfter | Alan Baker ⓘ |
| provedBy | Alan Baker ⓘ |
| provides | effective lower bounds for linear forms in logarithms ⓘ |
| relatedTo |
Baker–Wüstholz theorem
ⓘ
linked to:
Baker theorem on linear forms in logarithms
Gelfond–Schneider theorem ⓘ Matveev’s theorem on linear forms in logarithms ⓘ the theory of heights in Diophantine geometry ⓘ |
| status | fundamental tool in modern Diophantine analysis ⓘ |
| timePeriod | 1960s ⓘ |
| usedFor |
bounding integer solutions of S-unit equations
ⓘ
bounding integer solutions of Thue equations ⓘ bounding integer solutions of Thue–Mahler equations ⓘ bounding integer solutions of exponential Diophantine equations ⓘ effective finiteness results for Diophantine equations ⓘ effective results in Diophantine approximation ⓘ effective results on the Mordell equation ⓘ effective versions of Siegel’s theorem on integral points ⓘ results on Catalan-type equations ⓘ results on Pillai-type equations ⓘ results on perfect powers in recurrence sequences ⓘ results on the Lebesgue–Nagell equation ⓘ results on the Ramanujan–Nagell equation ⓘ |
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Subject: Baker theorem on linear forms in logarithms Description of subject: The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Baker theorem on linear forms in logarithms
linked to: Baker theorem on linear forms in logarithms
linked to: Baker theorem on linear forms in logarithms
Liouville's inequality in Diophantine approximation
→
relatedConcept
→
Baker's theory of linear forms in logarithms
ⓘ
linked to: Baker theorem on linear forms in logarithms