ALGEBRAIC AND GEOMETRIC PROPERTIES OF LATTICE WALKS WITH STEPS OF EQUAL LENGTH

作者：Kolodziejczyk Krzysztof^*; Salapata Rafal

来源：Bulletin of the Australian Mathematical Society, 2017, 95(2): 338-346.

DOI：10.1017/S0004972716000836

• lattice d-walk
• terminal set
• sublattice
• quotient group
• prime number
• sum of squares

A lattice walk with all steps having the same length d is called a d-walk. Denote by T-d the terminal set, that is, the set of all lattice points that can be reached from the origin by means of a d-walk. We examine some geometric and algebraic properties of the terminal set. After observing that (T-d; +) is a normal subgroup of the group (Z(N); +), we ask questions about the quotient group Z(N)/T-d and give the number of elements of Z(2)/T-d in terms of d. To establish this result, we use several consequences of Fermat's theorem about representations of prime numbers of the form 4k + 1 as the sum of two squares. One of the consequences is the fact, observed by Sierpinski, that every natural power of such a prime number has exactly one relatively prime representation. We provide explicit formulas for the relatively prime integers in this representation.

• 出版日期2017-4