Squares from Blocks of Consecutive Integers : a Problem of Erdős and Graham
نویسندگان
چکیده
In this paper, we construct, given an integer r ≥ 5, an infinite family of r non-overlapping blocks of five consecutive integers with the property that their product is always a perfect square. In this particular situation, this answers a question of Erdős and Graham in the negative.
منابع مشابه
On a Question of Erdős and Graham
In this note, we sharpen work of Ulas to provide what is, in some sense, the minimal counterexample to a “conjecture” of Erdős and Graham about square values of products of disjoint blocks of consecutive integers.
متن کاملThe Erdős-Heilbronn Problem for Finite Groups
Additive Number Theory can be best described as the study of sums of sets of integers. A simple example is given two subsets A and B of a set of integers, what facts can we determine about A + B where A + B := {a + b | a ∈ A and b ∈ B}? We will state a result regarding this example shortly. We note that a very familiar problem in Number Theory, namely Lagrange’s theorem that every nonnegative i...
متن کاملA Note on Erdős-straus and Erdős-graham Divisibility Problems
In this paper we are interested in two problems stated in the book of Erdős and Graham. The first problem was stated by Erdős and Straus in the following way: Let n ∈ N+ be fixed. Does there exist a positive integer k such that k ∏
متن کاملOn partitions into squares of distinct integers whose reciprocals sum to 1
In 1963, Graham [1] proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured [2, Section D11] that for any sufficiently large integer, it can be partitioned into squares of distinct positive integers whose reciprocals sum to 1. In this study, we establish the exact bound for existence of su...
متن کاملTao’s Resolution of the Erdős Discrepancy Problem
This article gives a simplified account of some of the ideas behind Tao’s resolution of the Erdős discrepancy problem. The Erdős discrepancy problem is an easily stated question about arbitrary functions f from the positive integers to ±1. It asks whether the signs ±1 can be arranged evenly over all subsequences of the form kj for a given k ∈ N and as j varies. Precisely, must it always be the ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011