Bibliographic Details
| Title: |
On irrationals with Lagrange value exactly 3. |
| Authors: |
Cao, Zhe1 (AUTHOR) zhecao@mail.nankai.edu.cn, Erazo, Harold2 (AUTHOR) harold.erazo@impa.br, Moreira, Carlos Gustavo2,3 (AUTHOR) gugu@impa.br |
| Source: |
Journal of Number Theory. May2026, Vol. 282, p147-173. 27p. |
| Subjects: |
Irrational numbers, Lagrange problem, Real numbers, Cardinal numbers, Mathematics, Number theory |
| Abstract: |
For c > 0 , let X c denote the set of x ∈ R ﹨ Q such that | x − p q | < 1 c q 2 has only finitely many rational solutions p q. It is a classical fact, known since the 1950s, that X c is uncountable for c > 3 and countable for c < 3. However, the cardinality of X 3 does not appear to be present in the literature. We prove that X 3 is uncountable. More generally, we show that for any n ∈ N ∪ { ∞ } , the set of x ∈ R ﹨ Q with Lagrange value exactly 3 and such that | x − p q | < 1 3 q 2 has exactly n rational solutions p q is also uncountable. For a video summary of this paper, please visit https://youtu.be/VyKB99-kVeY. [ABSTRACT FROM AUTHOR] |
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| Database: |
Engineering Source |
