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ALGEBRAIC AND GEOMETRIC PROPERTIES OF LATTICE WALKS WITH STEPS OF EQUAL LENGTH
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 95 / Issue 2 / April 2017
- Published online by Cambridge University Press:
- 02 November 2016, pp. 338-346
- Print publication:
- April 2017
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- Article
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A lattice walk with all steps having the same length
$d$ is called a 
$d$-walk. Denote by 
${\mathcal{T}}_{d}$ 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 
$({\mathcal{T}}_{d},+)$ is a normal subgroup of the group 
$(\mathbb{Z}^{N},+)$, we ask questions about the quotient group 
$\mathbb{Z}^{N}/{\mathcal{T}}_{d}$ and give the number of elements of 
$\mathbb{Z}^{2}/{\mathcal{T}}_{d}$ 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 Sierpiński, 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.
