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ON A CONJECTURE REGARDING THE SYMMETRIC DIFFERENCE OF CERTAIN SETS
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 10 October 2024, pp. 1-8
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Let n be a positive integer and
$\underline {n}=\{1,2,\ldots ,n\}$. A conjecture arising from certain polynomial near-ring codes states that if 
$k\geq 1$ and 
$a_{1},a_{2},\ldots ,a_{k}$ are distinct positive integers, then the symmetric difference 
$a_{1}\underline {n}\mathbin {\Delta }a_{2}\underline {n}\mathbin {\Delta }\cdots \mathbin {\Delta }a_{k}\underline {n}$ contains at least n elements. Here, 
$a_{i}\underline {n}=\{a_{i},2a_{i},\ldots ,na_{i}\}$ for each i. We prove this conjecture for arbitrary n and for 
$k=1,2,3$.
