A note on the congruence n p k m p k n m ( mod p r )

Romeo Meštrović

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 1, page 59-65
  • ISSN: 0011-4642

Abstract

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In the paper we discuss the following type congruences: n p k m p k m n ( mod p r ) , where p is a prime, n , m , k and r are various positive integers with n m 1 , k 1 and r 1 . Given positive integers k and r , denote by W ( k , r ) the set of all primes p such that the above congruence holds for every pair of integers n m 1 . Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W ( k , r ) and inclusion relations between them for various values k and r . In particular, we prove that W ( k + i , r ) = W ( k - 1 , r ) for all k 2 , i 0 and 3 r 3 k , and W ( k , r ) = W ( 1 , r ) for all 3 r 6 and k 2 . We also noticed that some of these properties may be used for computational purposes related to congruences given above.

How to cite

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Meštrović, Romeo. "A note on the congruence ${np^k\atopwithdelims ()mp^k} \equiv {n\atopwithdelims ()m} \hspace{4.44443pt}(\@mod \; p^r)$." Czechoslovak Mathematical Journal 62.1 (2012): 59-65. <http://eudml.org/doc/246182>.

@article{Meštrović2012,
abstract = {In the paper we discuss the following type congruences: \[ \biggl (\{np^k\atop mp^k\}\biggr ) \equiv \left(m \atop n\right) \hspace\{10.0pt\}(\@mod \; p^r), \] where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W(k,r)=W(1,r)$ for all $3\le r\le 6$ and $k\ge 2$. We also noticed that some of these properties may be used for computational purposes related to congruences given above.},
author = {Meštrović, Romeo},
journal = {Czechoslovak Mathematical Journal},
keywords = {congruence; prime powers; Lucas’ theorem; Wolstenholme prime; set $W(k,r)$; congruence; prime powers; Lucas' theorem; Wolstenholme prime; set },
language = {eng},
number = {1},
pages = {59-65},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the congruence $\{np^k\atopwithdelims ()mp^k\} \equiv \{n\atopwithdelims ()m\} \hspace\{4.44443pt\}(\@mod \; p^r)$},
url = {http://eudml.org/doc/246182},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Meštrović, Romeo
TI - A note on the congruence ${np^k\atopwithdelims ()mp^k} \equiv {n\atopwithdelims ()m} \hspace{4.44443pt}(\@mod \; p^r)$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 59
EP - 65
AB - In the paper we discuss the following type congruences: \[ \biggl ({np^k\atop mp^k}\biggr ) \equiv \left(m \atop n\right) \hspace{10.0pt}(\@mod \; p^r), \] where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W(k,r)=W(1,r)$ for all $3\le r\le 6$ and $k\ge 2$. We also noticed that some of these properties may be used for computational purposes related to congruences given above.
LA - eng
KW - congruence; prime powers; Lucas’ theorem; Wolstenholme prime; set $W(k,r)$; congruence; prime powers; Lucas' theorem; Wolstenholme prime; set
UR - http://eudml.org/doc/246182
ER -

References

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  4. Kazandzidis, G. S., Congruences on the binomial coefficients, Bull. Soc. Math. Grèce, N. Ser. 9 (1968), 1-12. (1968) Zbl0179.06601MR0265271
  5. Lucas, E., Sur les congruences des nombres eulériens et les coefficients différentiels des functions trigonométriques suivant un module premier, Bull. S. M. F. 6 (1878), 49-54 French. (1878) MR1503769
  6. McIntosh, R. J., On the converse of Wolstenholme's Theorem, Acta Arith. 71 (1995), 381-389. (1995) Zbl0829.11003MR1339137
  7. McIntosh, R. J., Roettger, E. L., 10.1090/S0025-5718-07-01955-2, Math. Comput. 76 (2007), 2087-2094. (2007) Zbl1139.11003MR2336284DOI10.1090/S0025-5718-07-01955-2
  8. Meštrović, R., A note on the congruence n d m d n m ( mod q ) , Am. Math. Mon. 116 (2009), 75-77. (2009) MR2478756
  9. Sun, Z.-W., Davis, D. M., 10.1090/S0002-9947-07-04236-5, Trans. Am. Math. Soc. 359 (2007), 5525-5553. (2007) Zbl1119.11016MR2327041DOI10.1090/S0002-9947-07-04236-5
  10. Zhao, J., 10.1016/j.jnt.2006.05.005, J. Number Theory 123 (2007), 18-26. (2007) MR2295427DOI10.1016/j.jnt.2006.05.005

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