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Number of solutions to a special type of unit equations in two unknowns, II.
- Source :
-
Research in Number Theory . 4/1/2024, Vol. 10 Issue 2, p1-41. 41p. - Publication Year :
- 2024
-
Abstract
- This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers a, b and c all greater than 1 there is at most one solution to the equation a x + b y = c z in positive integers x, y and z, except for specific cases. The fundamental result proves the conjecture under some congruence condition modulo c on a and b. As applications the conjecture is confirmed to be true if c takes some small values including the Fermat primes found so far, and in particular this provides an analytic proof of the celebrated theorem of Scott (J Number Theory 44(2):153-165, 1993) solving the conjecture for c = 2 in a purely algebraic manner. The method can be generalized for smaller modulus cases, and it turns out that the conjecture holds true for infinitely many specific values of c not being perfect powers. The main novelty is to apply a special type of the p-adic analogue to Baker's theory on linear forms in logarithms via a certain divisibility relation arising from the existence of two hypothetical solutions to the equation. The other tools include Baker's theory in the complex case and its non-Archimedean analogue for number fields together with various elementary arguments through rational and quadratic numbers, and extensive computation. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 25220160
- Volume :
- 10
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Research in Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 176384518
- Full Text :
- https://doi.org/10.1007/s40993-024-00524-7