### Algebraic proofs for geometric statements.

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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.

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$satisfies$d < 1.55·1072$and$b < 6.21·1035$when4a<b,whileforb<4aonehaseither$c = a + b + 2√(ab+1) and $d<1.96\xb7{10}^{53}$...

The triples $(x,y,z)=(1,{z}^{z}-1,z)$, $(x,y,z)=({z}^{z}-1,1,z)$, where $z\in \mathbb{N}$, 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.

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 $5.441\xb7{10}^{26}$ Diophantine quintuples.

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