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001-es BibID:	BIBFORM120613
035-os BibID:	(WoS)001195375200001 (Scopus)85189167134
Első szerző:	Miyazaki, Takafumi
Cím:	Number of solutions to a special type of unit equations in two unknowns, II / Takafumi Miyazaki, István Pink
Dátum:	2024
ISSN:	2522-0160 2363-9555
Megjegyzések:	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 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 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.
Tárgyszavak:	Természettudományok Matematika- és számítástudományok idegen nyelvű folyóiratközlemény külföldi lapban folyóiratcikk S-unit equation Purely exponential equation Baker's method Non-Archimedean valuation Fermat prime
Megjelenés:	Research in Number Theory. - 10 : 2 (2024), p. 1-41. -
További szerzők:	Pink István (1973-) (matematikus)
Pályázati támogatás:	ANN130909 OTKA K128088 OTKA
Internet cím:	Szerző által megadott URL DOI Intézményi repozitóriumban (DEA) tárolt változat
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