Generalised Frobenius numbers: geometry of upper bounds, Frobenius graphs and exact formulas for arithmetic sequences
Dilbak Haji Mohammed School of Mathematics Cardiff University Cardiff, South Wales, UK
This thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy
February 7, 2017 2 Dedication
This dissertation is expressly dedicated to the memory of my father, Haji mohammed Haji who left us with the most precious asset in life, knowledge. I know that he would be the happiest father in the world to know that his daughter has completed her PhD studies. I also dedicate my work to my lovely mother Adila Mousa for her support, encouragement, and constant love that have sustained me throughout my life.
I also dedicate this work and express my special thanks to all my family members, friends, and colleagues whose words of encouragement halped me to write this dissertation.
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Summary
t Given a positive integer vector a = (a1, a2 . . . , ak) with
1 < a1 < ··· < ak and gcd(a1, . . . , ak) = 1 .
The Frobenius number of the vector a,Fk(a), is the largest positive integer that cannot be k P represented as aixi, where x1, . . . , xk are nonnegative integers. We also consider a generalised i=1 Frobenius number, known in the literature as the s-Frobenius number, Fs(a1, a2, . . . , ak), which k P is defined to be the largest integer that cannot be represented as aixi in at least s distinct i=1 ways. The classical Frobenius number corresponds to the case s = 1.
The main result of the thesis is the new upper bound for the 2-Frobenius number,
!1/(k−1) (k − 1)! F (a , . . . , a ) ≤ F (a , . . . , a ) + 2 (a ··· a )1/(k−1) , (0.0.1) 2 1 k 1 1 k 2(k−1) 1 k k−1