Another Note on Baker’s Theorem

In 1962, Mahler defined a measure for integer polynomials in several variables as the logarithmic integral over the torus. Many results exist about the values taken by the measure but many unsolved problems remain. In one variable, it is possible to express the measure as an effective limit of Riemann sums. We show that the same is true in several variables, using a non-obvious parametrisation of the torus together with Baker's Theorem on linear forms in logarithms of algebraic numbers.

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A new approach to Baker's theorem on linear forms in logarithms II

Diophantine Approximation and Transcendence Theory - Lecture Notes in Mathematics ◽

10.1007/bfb0078710 ◽

1987 ◽

pp. 203-211 ◽

Cited By ~ 2

Author(s):

G. Wüstholz

Keyword(s):

Linear Forms In Logarithms ◽

Linear Forms ◽

New Approach ◽

Baker's Theorem

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A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING PRIMITIVE PYTHAGOREAN TRIPLES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000605 ◽

2016 ◽

Vol 95 (1) ◽

pp. 5-13 ◽

Cited By ~ 4

Author(s):

MOU-JIE DENG ◽

DONG-MING HUANG

Keyword(s):

Positive Integer ◽

Diophantine Equation ◽

Integer Solution ◽

Character Theory ◽

Positive Integer Solution ◽

Pythagorean Triples ◽

Pythagorean Triple ◽

Positive Integers ◽

Baker's Theorem

Let $a,b,c$ be a primitive Pythagorean triple and set $a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$, where $m$ and $n$ are positive integers with $m>n$, $\text{gcd}(m,n)=1$ and $m\not \equiv n~(\text{mod}~2)$. In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is $(x,y,z)=(2,2,2)$. We use biquadratic character theory to investigate the case with $(m,n)\equiv (2,3)~(\text{mod}~4)$. We show that Jeśmanowicz’ conjecture is true in this case if $m+n\not \equiv 1~(\text{mod}~16)$ or $y>1$. Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if $(m,n)\equiv (2,3)~(\text{mod}~4)$ and $n<100$.

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Some Applications of Baker’s Theorem

Transcendental Numbers ◽

10.1007/978-1-4939-0832-5_20 ◽

2014 ◽

pp. 101-109

Author(s):

M. Ram Murty ◽

Purusottam Rath

Keyword(s):

Baker's Theorem

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ON THE FACTORIZATION OF f(n) FOR f(x) IN ℤ[x]

International Journal of Number Theory ◽

10.1142/s179304211350022x ◽

2013 ◽

Vol 09 (05) ◽

pp. 1225-1236 ◽

Cited By ~ 4

Author(s):

SAMUEL S. GROSS ◽

ANDREW F. VINCENT

Keyword(s):

Upper Bounds ◽

Finite Set ◽

Baker's Theorem

Let S be a finite set of rational primes. For a non-zero integer n, define [Formula: see text], where |n|p is the usual p-adic norm of n. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for [n(n+1)⋯(n+k)]S for any integer k > 0. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for [n(n + 1)]S and [n2 + 7]S, where S = {2, 3} and S = {2}, respectively. We extend Stewart's theorem to prove effective upper bounds for [f(n)]S for an arbitrary f(x) in ℤ[x] having at least two distinct roots.

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