On the Occurrence of Perfect Squares Among Values of Certain Polynomial Products

Gurel, Erhan
We prove that the product of first n consecutive values of the polynomial P(k) = 4k(4) + 1 is a perfect square infinitely often whereas the product of first n consecutive values of the polynomial Q(k) = k(4) + 4 is a perfect square only for n = 2.


On the special values of monic polynomials of hypergeometric type
Taşeli, Hasan (Springer Science and Business Media LLC, 2008-01-01)
Special values of monic polynomials y(n)(s), with leading coefficients of unity, satisfying the equation of hypergeometric type
On the arithmetic exceptionality of polynomial mappings
Küçüksakallı, Ömer (2018-02-01)
In this note we prove that certain polynomial mappings P-g(k) (x) is an element of Z[x] in n-variables obtained from simple complex Lie algebras g of arbitrary rank n1, are exceptional.
On the Polynomial Multiplication in Chebyshev Form
Akleylek, Sedat; Cenk, Murat; Özbudak, Ferruh (2012-04-01)
We give an efficient multiplication method for polynomials in Chebyshev form. This multiplication method is different from the previous ones. Theoretically, we show that the number of multiplications is at least as good as Karatsuba-based algorithm. Moreover, using the proposed method, we improve the number of additions slightly. We remark that our method works efficiently for any N and it is easy to implement. To the best of our knowledge, the proposed method has the best multiplication and addition comple...
On the arithmetic complexity of Strassen-like matrix multiplications
Cenk, Murat (2017-05-01)
The Strassen algorithm for multiplying 2 x 2 matrices requires seven multiplications and 18 additions. The recursive use of this algorithm for matrices of dimension n yields a total arithmetic complexity of (7n(2.81) - 6n(2)) for n = 2(k). Winograd showed that using seven multiplications for this kind of matrix multiplication is optimal. Therefore, any algorithm for multiplying 2 x 2 matrices with seven multiplications is called a Strassen-like algorithm. Winograd also discovered an additively optimal Stras...
Value sets of folding polynomials over finite fields
Küçüksakallı, Ömer (2019-01-01)
Let k be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra g. We find the cardinality of the value sets of the folding polynomials P-g(k)(x) is an element of Z[x] of arbitrary rank n >= 1, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.
Citation Formats
E. Gurel, “On the Occurrence of Perfect Squares Among Values of Certain Polynomial Products,” AMERICAN MATHEMATICAL MONTHLY, pp. 597–599, 2016, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/63350.