Infinitely many distinct nonzero rational triples {a,b,c} exist making seven expressions (a+1 through abc+1) perfect squares, but none exist among positive integers.
Stoll, Diagonal genus 5 curves, elliptic curves overQ(t), and rational diophantine quin- tuples, Acta Arith.190(2019)
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Seven squares from three numbers
Infinitely many distinct nonzero rational triples {a,b,c} exist making seven expressions (a+1 through abc+1) perfect squares, but none exist among positive integers.