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US Patent No. 10855467
In a recent article, Nowicki introduced the concept of a special number. Specifically, an integer d is called special if for every integer m there exist solutions in non-zero integers a, b, c to the equation a2 +b9 −dc2 = m. In this article we investigate pairs of integers (n, d), with n ≥ 2, such that for every integer m there exist units a, b, an...
Let f (x) be a polynomial with non-negative integer coefficients for which f (10) is prime. A result of A. Cohn implies that if the coefficients of f (x) are ≤ 9, then f (x) is irreducible. In 1988, the first author showed that the bound 9 could be replaced by 10 30. We show here that the bound 9 can be replaced by the number in the title and that...
Let S be a finite set of rational primes. For a non-zero integer n, define , where |n|p is the usual p-adic norm of n. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for [n(n+1)⋯(n+k)]S for any integer k > 0. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for [n(...