OCTOGON MATHEMATICAL MAGAZINE Vol. 17, No.1, April 2009, pp 304-305
ISSN 1222-5657, ISBN 978-973-88255-5-0, www.hetfalu.ro/octogon
304
A note on Bang‘s and Zsigmond‘s theorems
J´ozsef S´andor38
ABSTRACT.In a recent note [2], an application of the so-called Birkhoff-Vandier theorem was given: We offer a history of this theorem, due to Bang and Zsygmondy.
Recently, in note [2], the following theorem of Zsygmondy from 1892 (see [4]) has been applied:
Theorem. Ifa, band nare integers witha > b >0,gdc(a, b) = 1 andn >2, then there is a prime divisorp ofan−bn such thatpis not a divisor ofak−bk for any integer with 1≤k < n, except for the casea= 2, b= 1, n= 6.
The caseb= 1 is due to Bang [1], who discovered it in 1886.
Both Bang‘s theorem and Zsigmond‘s theorem have been rediscovered many times in the XXth century. A partial list of references is given in [5], p. 361.
It should be noted that Zsygmond‘s theorem has itself been generalized to algebraic number fields. A list of references on generalizations of Zsigmond‘s theorem can be found in [3], from which earlier references may be obtained.
REFERENCES
[1]. Bang. A.S.,Taltheoretiske Undersogelser, Tidsskrifft Math., 5IV (1886), 70-80 and 130-137.
[2] Le. M., and Bencze,M.,An application of the Birkhoff-Vandiver theorem, Octogon Mathematical Magazine, Vol. 16, No. 2, October 2008, pp.
1357-1360.
[3] Stewart, C.L., On divisors of terms of linear recurrence sequences, J.
Reine Angen. Math., 333, (1982),pp. 12-31.
38Received: 04.02.2009
2000Mathematics Subject Classification. 11A25.
Key words and phrases. Primitive divisor‘s; Bang‘s theorem; Zsygmondy‘s theo- rem.
A note on Bang‘s and Zsigmond‘s theorems 305 [4] Zsigmondy, K., Zur Theorie der Potenzreste, Monatsch Math. Phys.
3(1982), pp. 265-284.
[5] Dandapat, G.G., Hunsucker, J.L., and Poerance, C., Some new results on odd perfect numbers, Pacific J. Math. 57(1975), pp. 359-364.
Babe¸s-Bolyai University,
Cluj and Miercurea Ciuc, Romania
