On sets of integers whose shifted products are powers
نویسنده
چکیده
Let N be a positive integer and let A be a subset of {1, . . . , N} with the property that aa′ + 1 is a pure power whenever a and a′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A| ≪ (logN)2/3(log logN)1/3.
منابع مشابه
On shifted products which are powers
Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj + 1 is a square for 1 ≤ i < j ≤ 4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grin...
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ورودعنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 115 شماره
صفحات -
تاریخ انتشار 2008