4 results
Supercongruences involving Motzkin numbers and central trinomial coefficients
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- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 03 October 2024, pp. 1-25
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Let Mn and Tn denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime
$p\ge 5$,
\begin{equation*}\begin{aligned}\\[-22pt]&\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2},\\[-6pt]\end{aligned}\end{equation*}where
$\left(-\right)$ is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.
APPLICATIONS OF CIRCULANT MATRICES TO DETERMINANTS INVOLVING
$\boldsymbol {k}$
TH POWER RESIDUES
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 106 / Issue 2 / October 2022
- Published online by Cambridge University Press:
- 09 February 2022, pp. 243-253
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- October 2022
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Sequences of low arithmetical complexity
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- Journal:
- RAIRO - Theoretical Informatics and Applications / Volume 40 / Issue 4 / October 2006
- Published online by Cambridge University Press:
- 08 November 2006, pp. 569-582
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- October 2006
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Almost Squares in Arithmetic Progression
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- Journal:
- Compositio Mathematica / Volume 138 / Issue 1 / August 2003
- Published online by Cambridge University Press:
- 04 December 2007, pp. 73-111
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- August 2003
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