On the ratio of two blocks of consecutive integers

1990 ◽  
Vol 100 (2) ◽  
Author(s):  
N Saradha ◽  
T N Shorey
Keyword(s):  
Consecutive Integers ◽  
Blocks Of Consecutive Integers

Products of disjoint blocks of integers being high powers

2019 ◽  
Vol 15 (01) ◽  
pp. 85-88
Author(s):  
M. Skałba
Keyword(s):  
Consecutive Integers ◽  
Blocks Of Consecutive Integers

Fix [Formula: see text] and put [Formula: see text]. Improving on results of [M. Skałba, Products of disjoint blocks of consecutive integers which are powers, Colloq. Math. 98 (2003) 1–3], we prove the following. For a given [Formula: see text] and each [Formula: see text], there exists [Formula: see text] such that for any exponent [Formula: see text], there exist nonnegative integers [Formula: see text] satisfying the equation [Formula: see text] and inequalities: [Formula: see text] and [Formula: see text] with [Formula: see text]

scholarly journals Blocks of consecutive integers in sumsets (A + B)t

2004 ◽  
Vol 70 (2) ◽  
pp. 283-291 ◽  
Author(s):  
Shu-Guang Guo ◽  
Yong-Gao Chen
Keyword(s):  
Arithmetic Progression ◽  
Consecutive Integers ◽  
Blocks Of Consecutive Integers ◽  
Ordered Pairs

Let A, B ⊆ {1, …, n}. For m ∈ Z, let rA,B(m) be the cardinality of the set of ordered pairs (a, b) ∈ A × B such that a + b = m. For t ≥ 1, denote by (A + B)t the set of the elements m for which rA, B(m) ≥ t. In this paper we prove that for any subsets A, B ⊆ {1, …, n} such that |A| + |B| ≥ (4n + 4t − 3)/3, the sumset (A + B)t contains a block of consecutive integers with the length at least |A| + |B| − 2t + 1, and that (a) for any two subsets A and B of {1, …, n} such that |A| + |B| ≥ (4n)/3, there exists an arithmetic progression of length n in A + B; (b) for any 2 ≤ r ≤ (4n − 1)/3, there exist two subsets A and B of {1, …, n} with |A| + |B| = r such that any arithmetic progression in A + B has the length at most (2n − 1)/3 + 1.

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.

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.

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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