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Dickson Polynomials that are Permutations

Cipu, Mihai — 2004

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 11T06, 13P10. A theorem of S.D. Cohen gives a characterization for Dickson polynomials of the second kind that permutes the elements of a finite field of cardinality the square of the characteristic. Here, a different proof is presented for this result. Research supported by the CERES program of the Ministry of Education, Research and Youth, contract nr. 39/2002.

Further remarks on Diophantine quintuples

Mihai Cipu — 2015

Acta Arithmetica

A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < e s a t i s f i e s d < 1.55·1072 a n d b < 6.21·1035 w h e n 4 a < b , w h i l e f o r b < 4 a o n e h a s e i t h e r c = a + b + 2√(ab+1) and d < 1 . 96 · 10 53 ...

Complete solution of the Diophantine equation x y + y x = z z

Mihai Cipu — 2019

Czechoslovak Mathematical Journal

The triples ( x , y , z ) = ( 1 , z z - 1 , z ) , ( x , y , z ) = ( z z - 1 , 1 , z ) , where z , satisfy the equation x y + y x = z z . In this paper it is shown that the same equation has no integer solution with min { x , y , z } > 1 , thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

Searching for Diophantine quintuples

Mihai CipuTim Trudgian — 2016

Acta Arithmetica

We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most 5 . 441 · 10 26 Diophantine quintuples.

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