Let S(α) be the set of all integers of the form αn 2 for n ≥ α −1/2 , where x denotes the integer part of x. We investigate the existence of tuples (k, , m) of integers such that all of k, , m, k + , + m, m + k, k + + m belong to S(α). Let T (α) be the set of all such tuples. In this article, we reveal that T (α) is infinite for all α ∈ (0, 1) ∩ Q and for almost all α ∈ (0, 1) in the sense of Lebesgue measure. As a corollary, we disclose that the cardinality of α ∈ (0, 1) such that #T (α) = 0 is at most countable. Furthermore, we show that if there exists α > 0 such that T (α) is finite, then there is no perfect Euler brick. We also study the set of all integers of the form αn 2 for n ∈ N.
Let p be a prime number. A chain {p, 2p + 1, 4p + 3, • • • , (p + 1)2 l(p)−1 − 1} is called the Cunningham chain generated by p if all elements are prime numbers and (p + 1)2 l(p) − 1 is composite. Then l(p) is called the length of the Cunningham chain. It is conjectured by Bateman and Horn in 1962 that the number of prime numbers p ≤ N such that l(p) ≥ k is asymptotically equal to B k N/(log N ) k with a real B k > 0 for all natural numbers k. This suggests that l(p) = Ω(log p/ log log p). However, so far no good estimation is known. It has not even been proven whether lim sup p→∞ l(p) is infinite or not. All we know is that l(p) = 5 if p = 2 and l(p) < p for odd p by Fermat's little theorem. Let α ≥ 3 be an integer. In this article, a generalized Fibonacci sequence Fα = {Fn} ∞ n=0 is defined as F 0 = 0, F 1 = 1, F n+2 = αF n+1 + Fn(n ≥ 0), and Fα σ (n) = d|n,0
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