Abstract
In this paper we prove that if $$\{a,b,c\}$$ is a Diophantine triple with $$a<b<c$$ , then $$\{a+1,b,c\}$$ cannot be a Diophantine triple. Moreover, we show that if $$\{a_1,b,c\}$$ and $$\{a_2,b,c\}$$ are Diophantine triples with $$a_1<a_2<b<c < 16b^3$$ , then $$\{a_1,a_2,b,c\}$$ is a Diophantine quadruple. In view of these results, we conjecture that if $$\{a_1,b,c\}$$ and $$\{a_2,b,c\}$$ are Diophantine triples with $$a_1<a_2<b<c$$ , then $$\{a_1,a_2,b,c\}$$ is a Diophantine quadruple.
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