Numerical Semigroups Generated by Squares and Cubes of Three Consecutive Integers

2019 ◽  
pp. 101-121
Author(s):  
Leonid G. Fel
Keyword(s):  
Numerical Semigroups ◽  
Consecutive Integers

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals Almost canonical ideals and GAS numerical semigroups

2021 ◽  
pp. 1-24
Author(s):  
Marco D’Anna ◽  
Francesco Strazzanti
Keyword(s):  
Numerical Semigroups

On triplets with descending largest prime factors

2001 ◽  
Vol 38 (1-4) ◽  
pp. 45-50 ◽  
Author(s):  
A. Balog
Keyword(s):  
Prime Factor ◽  
Prime Factors ◽  
Consecutive Integers

For an integer n≯1 letP(n) be the largest prime factor of n. We prove that there are infinitely many triplets of consecutive integers with descending largest prime factors, that is P(n - 1) ≯P(n)≯P(n+1) occurs for infinitely many integers n.

Numerical semigroups whose fractions are of maximal embedding dimension

2010 ◽  
Vol 82 (3) ◽  
pp. 412-422 ◽  
Author(s):  
David E. Dobbs ◽  
Harold J. Smith
Keyword(s):  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Maximal Embedding Dimension

scholarly journals The set of numerical semigroups of a given genus

2012 ◽  
Vol 85 (2) ◽  
pp. 255-267 ◽  
Author(s):  
V. Blanco ◽  
J. C. Rosales
Keyword(s):  
Numerical Semigroups

Note on Pairs of Consecutive Integers the Sum of whose Squares is an Integral Square

1902 ◽  
Vol 23 ◽  
pp. 264-267
Author(s):  
Thomas Muir
Keyword(s):  
Consecutive Integers

The solution of the problem of finding such pairs of integers is not a thing of yesterday, as may be seen by consulting Hutton's translation of Ozanam's Recreations, i. pp. 46–8 (1814).

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