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We consider the Brocard-Ramanujan type Diophantine equation P(z) = n! + m!, where P is a polynomial with rational coefficients. We show that the ABC Conjecture implies that this equation has only finitely many integer solutions when d ≥ 2 and .
Maciej Gawron. "A note on the Diophantine equation P(z) = n! + m!." Colloquium Mathematicae 131.1 (2013): 53-58. <http://eudml.org/doc/284136>.
@article{MaciejGawron2013, abstract = {We consider the Brocard-Ramanujan type Diophantine equation P(z) = n! + m!, where P is a polynomial with rational coefficients. We show that the ABC Conjecture implies that this equation has only finitely many integer solutions when d ≥ 2 and $P(z) = a_dz^\{d\} + a_\{d-3\}z^\{d-3\} + ⋯ + a₁x + a₀$.}, author = {Maciej Gawron}, journal = {Colloquium Mathematicae}, keywords = {brocard-Ramanujan type equation; ABC conjecture; Diophantine equation}, language = {eng}, number = {1}, pages = {53-58}, title = {A note on the Diophantine equation P(z) = n! + m!}, url = {http://eudml.org/doc/284136}, volume = {131}, year = {2013}, }
TY - JOUR AU - Maciej Gawron TI - A note on the Diophantine equation P(z) = n! + m! JO - Colloquium Mathematicae PY - 2013 VL - 131 IS - 1 SP - 53 EP - 58 AB - We consider the Brocard-Ramanujan type Diophantine equation P(z) = n! + m!, where P is a polynomial with rational coefficients. We show that the ABC Conjecture implies that this equation has only finitely many integer solutions when d ≥ 2 and $P(z) = a_dz^{d} + a_{d-3}z^{d-3} + ⋯ + a₁x + a₀$. LA - eng KW - brocard-Ramanujan type equation; ABC conjecture; Diophantine equation UR - http://eudml.org/doc/284136 ER -