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math.GM2026
On the polynomial values represented by binary quadratic forms
Bogdan Grechuk, Jamal Agbanwa
Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate integral binary quadratic form and a univariate polynomial with integer coe…
math.GM2026
On the shortest open cubic equations
Bogdan Grechuk, Ashleigh Ratcliffe
We use cubic reciprocity to prove that the equation has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of i…
math.GM2024
A systematic approach to Diophantine equations: open problems
Bogdan Grechuk
This paper collects polynomial Diophantine equations that are simple to state but apparently difficult to solve.