We provide an estimate for the number of nontrivial integer points on the Pellian surface $t^2 - du^2 = 1$ in a bounded region. We give a lower bound on the size of fundamental solutions for almost all $d$ in a certain class, based on a recent conjecture of Browning and Wilsch about integer points on log K3 surfaces. We also obtain an upper bound on the average of class number in this class, assuming the same conjecture.
Nous donnons une estimation du nombre de points entiers non triviaux sur la surface pellienne $t^2 - d u^2 = 1$ dans une région bornée. Nous établissons une borne inférieure pour la taille des solutions fondamentales pour presque tout $d$ appartenant à une certaine classe, en nous fondant sur une conjecture récente de Browning et Wilsch concernant les points entiers sur les surfaces log K3. Nous obtenons également une borne supérieure pour la moyenne du nombre de classes dans cette classe, sous la même hypothèse conjecturale.
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Keywords: Pell equation, log K3 surfaces
Yijie Diao 1
CC-BY-ND 4.0
@article{JTNB_2025__37_3_973_0,
author = {Yijie Diao},
title = {Class numbers and integer points on some {Pellian} surfaces},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {973--988},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {3},
doi = {10.5802/jtnb.1348},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1348/}
}
TY - JOUR AU - Yijie Diao TI - Class numbers and integer points on some Pellian surfaces JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 973 EP - 988 VL - 37 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1348/ DO - 10.5802/jtnb.1348 LA - en ID - JTNB_2025__37_3_973_0 ER -
%0 Journal Article %A Yijie Diao %T Class numbers and integer points on some Pellian surfaces %J Journal de théorie des nombres de Bordeaux %D 2025 %P 973-988 %V 37 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1348/ %R 10.5802/jtnb.1348 %G en %F JTNB_2025__37_3_973_0
Yijie Diao. Class numbers and integer points on some Pellian surfaces. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 973-988. doi: 10.5802/jtnb.1348
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