Geodesic
On the lonely runner conjecture
Mathematica Bohemica, Tome 135 (2010) no. 1, pp. 63-68

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
Suppose $k+1$ runners having nonzero distinct constant speeds run laps on a unit-length circular track. The Lonely Runner Conjecture states that there is a time at which a given runner is at distance at least $1/(k+1)$ from all the others. The conjecture has been already settled up to seven ($k \leq 6$) runners while it is open for eight or more runners. In this paper the conjecture has been verified for four or more runners having some particular speeds using elementary tools.
Suppose $k+1$ runners having nonzero distinct constant speeds run laps on a unit-length circular track. The Lonely Runner Conjecture states that there is a time at which a given runner is at distance at least $1/(k+1)$ from all the others. The conjecture has been already settled up to seven ($k \leq 6$) runners while it is open for eight or more runners. In this paper the conjecture has been verified for four or more runners having some particular speeds using elementary tools.
DOI : 10.21136/MB.2010.140683
Classification : 11B25, 11B75
Keywords: congruences; arithmetic progression; bi-arithmetic progression 
Pandey, Ram Krishna. On the lonely runner conjecture. Mathematica Bohemica, Tome 135 (2010) no. 1, pp. 63-68. doi: 10.21136/MB.2010.140683
@article{10_21136_MB_2010_140683,
     author = {Pandey, Ram Krishna},
     title = {On the lonely runner conjecture},
     journal = {Mathematica Bohemica},
     pages = {63--68},
     year = {2010},
     volume = {135},
     number = {1},
     doi = {10.21136/MB.2010.140683},
     mrnumber = {2643356},
     zbl = {1224.11013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140683/}
}
TY  - JOUR
AU  - Pandey, Ram Krishna
TI  - On the lonely runner conjecture
JO  - Mathematica Bohemica
PY  - 2010
SP  - 63
EP  - 68
VL  - 135
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140683/
DO  - 10.21136/MB.2010.140683
LA  - en
ID  - 10_21136_MB_2010_140683
ER  - 
%0 Journal Article
%A Pandey, Ram Krishna
%T On the lonely runner conjecture
%J Mathematica Bohemica
%D 2010
%P 63-68
%V 135
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140683/
%R 10.21136/MB.2010.140683
%G en
%F 10_21136_MB_2010_140683

[1] Barajas, J., Serra, O.: Regular chromatic number and the lonely runner problem. Electron. Notes Discrete Math. 29 (2007), 479-483. | DOI | MR | Zbl

[2] Barajas, J., Serra, O.: The lonely runner with seven runners. Electron. J. Combin. 15 (2008), \# R 48. | DOI | MR | Zbl

[3] Betke, U., Wills, J. M.: Untere Schranken für zwei diophantische Approximations-Funktionen. Monatsh. Math. 76 (1972), 214-217. | DOI | MR | Zbl

[4] Bienia, W., Goddyn, L., Gvozdjak, P., Sebő, A., Tarsi, M.: Flows, view obstructions and the lonely runner. J. Combin. Theory Ser. B 72 (1998), 1-9. | DOI | MR

[5] Bohman, T., Holzman, R., Kleitman, D.: Six lonely runners. Electron. J. Combin. 8 (2001), \# R 3. | DOI | MR | Zbl

[6] Cusick, T. W.: View-obstruction problems in $n$-dimensional geometry. J. Combin. Theory Ser. A 16 (1974), 1-11. | DOI | MR | Zbl

[7] Cusick, T. W., Pomerance, C.: View-obstruction problems III. J. Number Theory 19 (1984), 131-139. | DOI | MR | Zbl

[8] Freiman, G. A.: Foundations of a structural theory of set addition. Transl. Math. Monogr. 37 (1973), American Mathematical Society, Providence, R.I | MR | Zbl

[9] Freiman, G. A.: Inverse problem of additive number theory IV. On addition of finite sets II. Ucen. Zap. Elabuz. Gos. Ped. Inst. 8 (1960), 72-116.

[10] Haralambis, N. M.: Sets of integers with missing differences. J. Combin. Theory Ser. A 23 (1977), 22-33. | DOI | MR | Zbl

[11] Jin, R.: Freiman's inverse problem with small doubling property. Adv. Math. 216 (2007), 711-752. | DOI | MR | Zbl

[12] Pandey, R. K.: A note on the lonely runner conjecture. J. Integer Sequences 12 (2009), Article 09.4.6. | MR | Zbl

[13] Renault, J.: View-obstruction: a shorter proof for six lonely runners. Discrete Math. 287 (2004), 93-101. | DOI | MR

[14] Wills, J. M.: Zwei Sätze über inhomogene diophantische Approximation von Irrationalzahlen. Monatsh. Math. 71 (1967), 263-269. | DOI | MR | Zbl

Cité par Sources :