On factorials which are products of factorials

@article{Luca2007OnFW,
  title={On factorials which are products of factorials},
  author={Florian Luca},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  year={2007},
  volume={143},
  pages={533 - 542}
}
  • F. Luca
  • Published 1 November 2007
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
Abstract In this paper, we look at the Diophantine equation (0.1)$ n!=\prod_{i=1}^t a_i!\qquad n>a_1\ge a_2\ge \cdots\ge a_t\ge 2. $ Under the ABC conjecture, we show that it has only finitely many nontrivial solutions. Unconditionally, we show that the set of n for which the above equation admits an integer solution a1,. . .,at is of asymptotic density zero. 
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References

SHOWING 1-10 OF 14 REFERENCES
On polynomial-factorial diophantine equations
We study equations of the form P(x) = n! and show that for some classes of polynomials P the equation has only finitely many solutions. This is the case, say, if P is irreducible (of degree greater
The product of consecutive integers is never a power
has no solution in integers with k >_ 2, 1 >_ 2 and n >_ 0 . (These restrictions on k, 1 and n will be implicit throughout this paper .) The early literature on this subject can be found in Dickson's
Number of prime divisors in a product of consecutive integers
For an integer ν > 1, we denote by ω(ν) and P (ν) the number of distinct prime divisors of ν and the greatest prime factor of ν, respectively, and let ω(1) = 0, P (1) = 1. Also we denote by W (∆′)
On the equation P(x) = n! and a question of Erdős
SOME PROBLEMS AND RESULTS ON THE IRRATIONALITY OF THE SUM OF INFINITE SERIES by
It is usually extremely difficult to decide whether the sum of a conver-gent infinite series is irrational or not. 2'1 was proved by Euler to be a polynomial in a and is thus transcendental, but kE
An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers
In this paper we study linear forms with rational integer coefficients (, ), where the are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an
A uniqueness theorem for a generalized solution of a system of two quasilinear equations in the class of bounded, measurable functions
A theorem is formulated and proved regarding the uniqueness of a generalized solution of Cauchy's for a hyperbolic system consisting of two first-order quasilinear equations with one spatial
Inverse piezoelectric effect and electrostrictive effect in polarized poly(vinylidene fluoride) films
The piezoelectric stress coefficient and electrostriction coefficient for unpoled and poled films of polyvinylidene fluoride have been determined by applying a sinusoidal electric field and detecting
The Diophantine Equation n! + 1 = m2
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