The Alladi–Grinstead constant is a mathematical constant that arises in the study of the factorial function, and in particular the decomposition of factorials into prime numbers. It is named for the mathematicians Krishnaswami Alladi and Charles Grinstead. The decimal representation of the Alladi–Grinstead constant begins,
0.80939402054063913071793188059409131721595399242500030424202871504... (sequence A085291 in the OEIS).
The exact value of the constant can be written as the exponential of a certain infinite series expressed using the Riemann zeta function. The sequence to which the Alladi–Grinstead constant is the limit arises in the investigation of how to write n factorial as the product of n large factors, a topic studied by Paul Erdős and others.
The factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers. For example, the factorial of 3 is
As a less trivial example:
where the are prime numbers and the are integers greater than or equal to 1. Let
As tends to infinity, approaches a limiting value, the Alladi–Grinstead constant. This constant can be written as an exponential:
- Mathematical constants
- Golomb–Dickman constant
- Weisstein, Eric. "Alladi-Grinstead Constant". mathworld.wolfram.com. Retrieved 2017-05-01.
- Alladi, Krishnaswami; Grinstead, Charles. "On the decomposition of n! into prime powers". Journal of Number Theory. 9 (4): 452–458. doi:10.1016/0022-314x(77)90006-3.
- Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 120–122. ISBN 978-0521818056. Search this book on
- Guy, Richard K. (1994). "Factorial n as the Product of n Large Factors". Unsolved Problems in Number Theory. Springer-Verlag. p. 79. ISBN 978-0387208602. Search this book on
- Weisstein, Eric. "Convergence Improvement". mathworld.wolfram.com. Retrieved 2017-05-03.
- (sequence A085290 in the OEIS)
- (sequence A085288 in the OEIS)
- (sequence A085289 in the OEIS)
- (sequence A085291 in the OEIS)
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