On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski. "On the Brocard-Ramanujan problem and generalizations." Colloquium Mathematicae 126.1 (2012): 105-110. <http://eudml.org/doc/284217>.

@article{AndrzejDąbrowski2012,
abstract = {Let $p_i$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n² - 1 = p₁^\{α₁\} ⋯ p_k^\{α_k\}$ in positive integers n and α₁,..., $α_k$. This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).},
author = {Andrzej Dąbrowski},
journal = {Colloquium Mathematicae},
keywords = {Diophantine equations; ABC conjecture},
language = {eng},
number = {1},
pages = {105-110},
title = {On the Brocard-Ramanujan problem and generalizations},
url = {http://eudml.org/doc/284217},
volume = {126},
year = {2012},
}

TY - JOUR
AU - Andrzej Dąbrowski
TI - On the Brocard-Ramanujan problem and generalizations
JO - Colloquium Mathematicae
PY - 2012
VL - 126
IS - 1
SP - 105
EP - 110
AB - Let $p_i$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n² - 1 = p₁^{α₁} ⋯ p_k^{α_k}$ in positive integers n and α₁,..., $α_k$. This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).
LA - eng
KW - Diophantine equations; ABC conjecture
UR - http://eudml.org/doc/284217
ER -